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case inl
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
x : E
β’ (Finset.sup β
p) x = 0 β¨ β i β β
, (Finset.sup β
p) x = (p i) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· | left | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case inl.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
x : E
β’ (Finset.sup β
p) x = 0 | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; | rfl | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
hs : Finset.Nonempty s
β’ (Finset.sup s p) x = 0 β¨ β i β s, (Finset.sup s p) x = (p i) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· | right | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
case inr.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
hs : Finset.Nonempty s
β’ β i β s, (Finset.sup s p) x = (p i) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; | exact exists_apply_eq_finset_sup p hs x | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; | Mathlib.Analysis.Seminorm.408_0.ywwMCgoKeIFKDZ3 | theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
C : ββ₯0
β’ Finset.sup s (C β’ p) = C β’ Finset.sup s p | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
| ext x | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
C : ββ₯0
x : E
β’ (Finset.sup s (C β’ p)) x = (C β’ Finset.sup s p) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
| rw [smul_apply, finset_sup_apply, finset_sup_apply] | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
C : ββ₯0
x : E
β’ β(Finset.sup s fun i => { val := ((C β’ p) i) x, property := (_ : 0 β€ ((C β’ p) i) x) }) =
C β’ β(Finset.sup s fun i => { val := (p i) x, property := (_ : 0 β€ (p i) x) }) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
| symm | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
C : ββ₯0
x : E
β’ C β’ β(Finset.sup s fun i => { val := (p i) x, property := (_ : 0 β€ (p i) x) }) =
β(Finset.sup s fun i => { val := ((C β’ p) i) x, property := (_ : 0 β€ ((C β’ p) i) x) }) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
| exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©)) | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
| Mathlib.Analysis.Seminorm.414_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
β’ Finset.sup s p β€ β i in s, p i | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
| classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
β’ Finset.sup s p β€ β i in s, p i | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
| refine' Finset.sup_le_iff.mpr _ | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
β’ β b β s, p b β€ β i in s, p i | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
| intro i hi | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
i : ΞΉ
hi : i β s
β’ p i β€ β i in s, p i | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
| rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left] | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
i : ΞΉ
hi : i β s
β’ 0 β€ β x in s \ {i}, p x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
| exact bot_le | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
| Mathlib.Analysis.Seminorm.421_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
a : β
ha : 0 β€ a
h : β i β s, (p i) x β€ a
β’ (Finset.sup s p) x β€ a | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
| lift a to ββ₯0 using ha | theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
| Mathlib.Analysis.Seminorm.429_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
a : ββ₯0
h : β i β s, (p i) x β€ βa
β’ (Finset.sup s p) x β€ βa | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
| rw [finset_sup_apply, NNReal.coe_le_coe] | theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
| Mathlib.Analysis.Seminorm.429_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
a : ββ₯0
h : β i β s, (p i) x β€ βa
β’ (Finset.sup s fun i => { val := (p i) x, property := (_ : 0 β€ (p i) x) }) β€ a | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
| exact Finset.sup_le h | theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
| Mathlib.Analysis.Seminorm.429_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
a : β
ha : 0 < a
h : β i β s, (p i) x < a
β’ (Finset.sup s p) x < a | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
| lift a to ββ₯0 using ha.le | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
| Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
a : ββ₯0
ha : 0 < βa
h : β i β s, (p i) x < βa
β’ (Finset.sup s p) x < βa | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
| rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff] | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
| Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
a : ββ₯0
ha : 0 < βa
h : β i β s, (p i) x < βa
β’ β b β s, { val := (p b) x, property := (_ : 0 β€ (p b) x) } < a | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· | exact h | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· | Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒΉβΈ : SeminormedRing π
instβΒΉβ· : SeminormedRing πβ
instβΒΉβΆ : SeminormedRing πβ
Οββ : π β+* πβ
instβΒΉβ΅ : RingHomIsometric Οββ
Οββ : πβ β+* πβ
instβΒΉβ΄ : RingHomIsometric Οββ
Οββ : π β+* πβ
instβΒΉΒ³ : RingHomIsometric Οββ
instβΒΉΒ² : AddCommGroup E
instβΒΉΒΉ : AddCommGroup Eβ
instβΒΉβ° : AddCommGroup Eβ
instββΉ : AddCommGroup F
instββΈ : AddCommGroup G
instββ· : Module π E
instββΆ : Module πβ Eβ
instββ΅ : Module πβ Eβ
instββ΄ : Module π F
instβΒ³ : Module π G
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
x : E
a : ββ₯0
ha : 0 < βa
h : β i β s, (p i) x < βa
β’ β₯ < a | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· | exact NNReal.coe_pos.mpr ha | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· | Mathlib.Analysis.Seminorm.440_0.ywwMCgoKeIFKDZ3 | theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instββΆ : SeminormedRing π
instββ΅ : SeminormedCommRing πβ
Οββ : π β+* πβ
instββ΄ : RingHomIsometric Οββ
instβΒ³ : AddCommGroup E
instβΒ² : AddCommGroup Eβ
instβΒΉ : Module π E
instβ : Module πβ Eβ
p : Seminorm πβ Eβ
f : E βββ[Οββ] Eβ
c : πβ
xβ : E
β’ (comp p (c β’ f)) xβ = (βcββ β’ comp p f) xβ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
| rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply] | theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
| Mathlib.Analysis.Seminorm.464_0.ywwMCgoKeIFKDZ3 | theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x : E
β’ 0 β lowerBounds (range fun u => p u + q (x - u)) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
| rintro _ β¨x, rflβ© | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
| Mathlib.Analysis.Seminorm.482_0.ywwMCgoKeIFKDZ3 | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ x : E
β’ 0 β€ (fun u => p u + q (xβ - u)) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
| dsimp | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
| Mathlib.Analysis.Seminorm.482_0.ywwMCgoKeIFKDZ3 | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ x : E
β’ 0 β€ p x + q (xβ - x) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; | positivity | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; | Mathlib.Analysis.Seminorm.482_0.ywwMCgoKeIFKDZ3 | /-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
x : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
β’ β (a : π) (x : E),
AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
(a β’ x) =
βaβ *
AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
| intro a x | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
a : π
x : E
β’ AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
(a β’ x) =
βaβ *
AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
| obtain rfl | ha := eq_or_ne a 0 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
x : E
β’ AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
(0 β’ x) =
β0β *
AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· | rw [norm_zero, zero_mul, zero_smul] | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· | Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
case inl
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
x : E
β’ AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
0 =
0 | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
| refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β© | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβΒΉ : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
xβ : E
x : β
hx : 0 < x
β’ p 0 + q (0 - 0) < x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by | rwa [map_zero, sub_zero, map_zero, add_zero] | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by | Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
a : π
x : E
ha : a β 0
β’ AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
(a β’ x) =
βaβ *
AddGroupSeminorm.toFun
{ toFun := fun x => β¨
u, p u + q (x - u), map_zero' := (_ : AddGroupSeminorm.toFun srcβ 0 = 0),
add_le' :=
(_ :
β (r s : E),
AddGroupSeminorm.toFun srcβ (r + s) β€ AddGroupSeminorm.toFun srcβ r + AddGroupSeminorm.toFun srcβ s),
neg' := (_ : β (r : E), AddGroupSeminorm.toFun srcβ (-r) = AddGroupSeminorm.toFun srcβ r) }
x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
| simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub] | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
a : π
x : E
ha : a β 0
β’ β¨
u, p u + q (a β’ x - u) = β¨
i, p (a β’ i) + q (a β’ x - a β’ i) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
| refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _ | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
case inr
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
p q : Seminorm π E
srcβ : AddGroupSeminorm E := p.toAddGroupSeminorm β q.toAddGroupSeminorm
a : π
x : E
ha : a β 0
u : E
β’ p (a β’ (fun x => aβ»ΒΉ β’ x) u) + q (a β’ x - a β’ (fun x => aβ»ΒΉ β’ x) u) = p u + q (a β’ x - u) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
| rw [smul_inv_smulβ ha] | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
| Mathlib.Analysis.Seminorm.489_0.ywwMCgoKeIFKDZ3 | noncomputable instance instInf : Inf (Seminorm π E) where
inf p q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
srcβ : SemilatticeSup (Seminorm π E) := instSemilatticeSup
p q : Seminorm π E
x : E
β’ p x + q (x - x) β€ (fun f => βf) p x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
| simp only [sub_self, map_zero, add_zero] | noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
| Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm π E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
srcβ : SemilatticeSup (Seminorm π E) := instSemilatticeSup
p q : Seminorm π E
x : E
β’ p x β€ p x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; | rfl | noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; | Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm π E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
srcβ : SemilatticeSup (Seminorm π E) := instSemilatticeSup
p q : Seminorm π E
x : E
β’ p 0 + q (x - 0) β€ (fun f => βf) q x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
| simp only [sub_self, map_zero, zero_add, sub_zero] | noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
| Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm π E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
srcβ : SemilatticeSup (Seminorm π E) := instSemilatticeSup
p q : Seminorm π E
x : E
β’ q x β€ q x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; | rfl | noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; | Mathlib.Analysis.Seminorm.514_0.ywwMCgoKeIFKDZ3 | noncomputable instance instLattice : Lattice (Seminorm π E) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instββ΅ : NormedField π
instββ΄ : AddCommGroup E
instβΒ³ : Module π E
pβ qβ : Seminorm π E
x : E
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
r : R
p q : Seminorm π E
β’ r β’ (p β q) = r β’ p β r β’ q | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
| ext | theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
| Mathlib.Analysis.Seminorm.526_0.ywwMCgoKeIFKDZ3 | theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instββ΅ : NormedField π
instββ΄ : AddCommGroup E
instβΒ³ : Module π E
pβ qβ : Seminorm π E
x : E
instβΒ² : SMul R β
instβΒΉ : SMul R ββ₯0
instβ : IsScalarTower R ββ₯0 β
r : R
p q : Seminorm π E
xβ : E
β’ (r β’ (p β q)) xβ = (r β’ p β r β’ q) xβ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
| simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add] | theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
| Mathlib.Analysis.Seminorm.526_0.ywwMCgoKeIFKDZ3 | theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
β’ iSup (fun p => ββp) 0 = 0 | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
| rw [iSup_apply, β @Real.ciSup_const_zero s] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
β’ β¨ i, βi 0 = β¨ x, 0 | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
| congr! | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
xβ : βs
β’ βxβ 0 = 0 | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
| rename_i _ _ _ i | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
i : βs
β’ βi 0 = 0 | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
| exact map_zero i.1 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
x y : E
β’ iSup (fun p => ββp) (x + y) β€ iSup (fun p => ββp) x + iSup (fun p => ββp) y | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
| rcases h with β¨q, hqβ© | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
β’ iSup (fun p => ββp) (x + y) β€ iSup (fun p => ββp) x + iSup (fun p => ββp) y | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
| obtain rfl | h := s.eq_empty_or_nonempty | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inl
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' β
)
β’ iSup (fun p => ββp) (x + y) β€ iSup (fun p => ββp) x + iSup (fun p => ββp) y | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· | simp [Real.ciSup_empty] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inr
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
h : Set.Nonempty s
β’ iSup (fun p => ββp) (x + y) β€ iSup (fun p => ββp) x + iSup (fun p => ββp) y | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
| haveI : Nonempty βs := h.coe_sort | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inr
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
h : Set.Nonempty s
this : Nonempty βs
β’ iSup (fun p => ββp) (x + y) β€ iSup (fun p => ββp) x + iSup (fun p => ββp) y | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
| simp only [iSup_apply] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inr
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
h : Set.Nonempty s
this : Nonempty βs
β’ β¨ i, βi (x + y) β€ (β¨ i, βi x) + β¨ i, βi y | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
| refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i) | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_1
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
h : Set.Nonempty s
this : Nonempty βs
i : βs
β’ q x β upperBounds (range fun i => AddGroupSeminorm.toFun (βi).toAddGroupSeminorm x) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> | rw [mem_upperBounds, forall_range_iff] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_2
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
h : Set.Nonempty s
this : Nonempty βs
i : βs
β’ q y β upperBounds (range fun i => AddGroupSeminorm.toFun (βi).toAddGroupSeminorm y) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> | rw [mem_upperBounds, forall_range_iff] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_1
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
h : Set.Nonempty s
this : Nonempty βs
i : βs
β’ β (i : { x // x β s }), AddGroupSeminorm.toFun (βi).toAddGroupSeminorm x β€ q x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> | exact fun j => hq (mem_image_of_mem _ j.2) _ | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case intro.inr.refine'_2
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
x y : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
h : Set.Nonempty s
this : Nonempty βs
i : βs
β’ β (i : { x // x β s }), AddGroupSeminorm.toFun (βi).toAddGroupSeminorm y β€ q y | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> | exact fun j => hq (mem_image_of_mem _ j.2) _ | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> | Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
x : E
β’ iSup (fun p => ββp) (-x) = iSup (fun p => ββp) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
| simp only [iSup_apply] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
x : E
β’ β¨ i, βi (-x) = β¨ i, βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
| congr! 2 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβΒΉ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
x : E
xβ : βs
β’ βxβ (-x) = βxβ x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
| rename_i _ _ _ i | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
x : E
i : βs
β’ βi (-x) = βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
| exact i.1.neg' _ | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
a : π
x : E
β’ AddGroupSeminorm.toFun
{ toFun := β¨ p, ββp, map_zero' := (_ : iSup (fun p => ββp) 0 = 0),
add_le' := (_ : β (x y : E), iSup (fun p => ββp) (x + y) β€ iSup (fun p => ββp) x + iSup (fun p => ββp) y),
neg' := (_ : β (x : E), iSup (fun p => ββp) (-x) = iSup (fun p => ββp) x) }
(a β’ x) =
βaβ *
AddGroupSeminorm.toFun
{ toFun := β¨ p, ββp, map_zero' := (_ : iSup (fun p => ββp) 0 = 0),
add_le' := (_ : β (x y : E), iSup (fun p => ββp) (x + y) β€ iSup (fun p => ββp) x + iSup (fun p => ββp) y),
neg' := (_ : β (x : E), iSup (fun p => ββp) (-x) = iSup (fun p => ββp) x) }
x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
| simp only [iSup_apply] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
a : π
x : E
β’ β¨ i, βi (a β’ x) = βaβ * β¨ i, βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
| rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x] | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
a : π
x : E
β’ β¨ i, βi (a β’ x) = β¨ i, βaβ β’ βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
| congr! | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβΒΉ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
a : π
x : E
xβ : βs
β’ βxβ (a β’ x) = βaβ β’ βxβ x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
| rename_i _ _ _ i | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
case h.e'_4.h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
h : BddAbove (FunLike.coe '' s)
a : π
x : E
i : βs
β’ βi (a β’ x) = βaβ β’ βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
| exact i.1.smul' a x | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
| Mathlib.Analysis.Seminorm.537_0.ywwMCgoKeIFKDZ3 | /-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
H : BddAbove (FunLike.coe '' s)
p : Seminorm π E
hp : p β s
x : E
β’ (fun f => βf) p x β€ (fun f => βf) (sSup s) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
| dsimp | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
H : BddAbove (FunLike.coe '' s)
p : Seminorm π E
hp : p β s
x : E
β’ p x β€ (sSup s) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
| rw [Seminorm.coe_sSup_eq' H, iSup_apply] | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
H : BddAbove (FunLike.coe '' s)
p : Seminorm π E
hp : p β s
x : E
β’ p x β€ β¨ i, βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
| rcases H with β¨q, hqβ© | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) | Mathlib_Analysis_Seminorm |
case intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
p : Seminorm π E
hp : p β s
x : E
q : E β β
hq : q β upperBounds (FunLike.coe '' s)
β’ p x β€ β¨ i, βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
| exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ© | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
| Mathlib.Analysis.Seminorm.595_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
x : E
p : ΞΉ β Seminorm π E
β’ BddAbove (range p) β β (x : E), BddAbove (range fun i => (p i) x) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
| rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi] | protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
| Mathlib.Analysis.Seminorm.606_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
x : E
p : ΞΉ β Seminorm π E
β’ (β (a : E), BddAbove (range fun i => (FunLike.coe β p) i a)) β β (x : E), BddAbove (range fun i => (p i) x) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; | rfl | protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; | Mathlib.Analysis.Seminorm.606_0.ywwMCgoKeIFKDZ3 | protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉβ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
x : E
ΞΉ : Type u_13
p : ΞΉ β Seminorm π E
hp : BddAbove (range p)
β’ β(β¨ i, p i) = β¨ i, β(p i) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
| rw [β sSup_range, Seminorm.coe_sSup_eq hp] | protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
| Mathlib.Analysis.Seminorm.615_0.ywwMCgoKeIFKDZ3 | protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉβ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
x : E
ΞΉ : Type u_13
p : ΞΉ β Seminorm π E
hp : BddAbove (range p)
β’ β¨ p_1, ββp_1 = β¨ i, β(p i) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
| exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p | protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
| Mathlib.Analysis.Seminorm.615_0.ywwMCgoKeIFKDZ3 | protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hp : BddAbove s
x : E
β’ (sSup s) x = β¨ p, βp x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
| rw [Seminorm.coe_sSup_eq hp, iSup_apply] | protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
| Mathlib.Analysis.Seminorm.621_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉβ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
ΞΉ : Type u_13
p : ΞΉ β Seminorm π E
hp : BddAbove (range p)
x : E
β’ (β¨ i, p i) x = β¨ i, (p i) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
| rw [Seminorm.coe_iSup_eq hp, iSup_apply] | protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
| Mathlib.Analysis.Seminorm.625_0.ywwMCgoKeIFKDZ3 | protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x : E
β’ sSup β
= β₯ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
| ext | protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
| Mathlib.Analysis.Seminorm.629_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x xβ : E
β’ (sSup β
) xβ = β₯ xβ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
| rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty] | protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
| Mathlib.Analysis.Seminorm.629_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x xβ : E
β’ 0 = β₯ xβ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
| rfl | protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
| Mathlib.Analysis.Seminorm.629_0.ywwMCgoKeIFKDZ3 | protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
p q : Seminorm π E
x : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
β’ IsLUB s (sSup s) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
| refine' β¨fun p hp x => _, fun p hp x => _β© | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β s
x : E
β’ (fun f => βf) p x β€ (fun f => βf) (sSup s) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> | haveI : Nonempty βs := hsβ.coe_sort | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β upperBounds s
x : E
β’ (fun f => βf) (sSup s) x β€ (fun f => βf) p x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> | haveI : Nonempty βs := hsβ.coe_sort | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β s
x : E
this : Nonempty βs
β’ (fun f => βf) p x β€ (fun f => βf) (sSup s) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
| dsimp | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β upperBounds s
x : E
this : Nonempty βs
β’ (fun f => βf) (sSup s) x β€ (fun f => βf) p x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
| dsimp | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β s
x : E
this : Nonempty βs
β’ p x β€ (sSup s) x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> | rw [Seminorm.coe_sSup_eq hsβ, iSup_apply] | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β upperBounds s
x : E
this : Nonempty βs
β’ (sSup s) x β€ p x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> | rw [Seminorm.coe_sSup_eq hsβ, iSup_apply] | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β s
x : E
this : Nonempty βs
β’ p x β€ β¨ i, βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· | rcases hsβ with β¨q, hqβ© | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_1.intro
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ qβ : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β s
x : E
this : Nonempty βs
q : Seminorm π E
hq : q β upperBounds s
β’ p x β€ β¨ i, βi x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
| exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ© | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
| Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
case refine'_2
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : NormedField π
instβΒΉ : AddCommGroup E
instβ : Module π E
pβ q : Seminorm π E
xβ : E
s : Set (Seminorm π E)
hsβ : BddAbove s
hsβ : Set.Nonempty s
p : Seminorm π E
hp : p β upperBounds s
x : E
this : Nonempty βs
β’ β¨ i, βi x β€ p x | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· | exact ciSup_le fun q => hp q.2 x | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· | Mathlib.Analysis.Seminorm.634_0.ywwMCgoKeIFKDZ3 | private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
x y : E
r : β
hr : 0 < r
β’ x β ball p x r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by | simp [hr] | theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by | Mathlib.Analysis.Seminorm.696_0.ywwMCgoKeIFKDZ3 | theorem mem_ball_self (hr : 0 < r) : x β ball p x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
x y : E
r : β
hr : 0 β€ r
β’ x β closedBall p x r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by | simp [hr] | theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by | Mathlib.Analysis.Seminorm.699_0.ywwMCgoKeIFKDZ3 | theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
x y : E
r : β
β’ y β ball p 0 r β p y < r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by | rw [mem_ball, sub_zero] | theorem mem_ball_zero : y β ball p 0 r β p y < r := by | Mathlib.Analysis.Seminorm.702_0.ywwMCgoKeIFKDZ3 | theorem mem_ball_zero : y β ball p 0 r β p y < r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
x y : E
r : β
β’ y β closedBall p 0 r β p y β€ r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by | rw [mem_closedBall, sub_zero] | theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by | Mathlib.Analysis.Seminorm.705_0.ywwMCgoKeIFKDZ3 | theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
xβ y : E
rβ : β
x : E
r : β
β’ closedBall p x r = β Ο, β (_ : Ο > r), ball p x Ο | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
| ext y | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
| Mathlib.Analysis.Seminorm.720_0.ywwMCgoKeIFKDZ3 | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
xβ yβ : E
rβ : β
x : E
r : β
y : E
β’ y β closedBall p x r β y β β Ο, β (_ : Ο > r), ball p x Ο | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; | simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le'] | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; | Mathlib.Analysis.Seminorm.720_0.ywwMCgoKeIFKDZ3 | theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
xβ y : E
r : β
x : E
hr : 0 < r
β’ ball 0 x r = univ | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
| rw [Set.eq_univ_iff_forall, ball] | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
| Mathlib.Analysis.Seminorm.724_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
p : Seminorm π E
xβ y : E
r : β
x : E
hr : 0 < r
β’ β (x_1 : E), x_1 β {y | 0 (y - x) < r} | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
| simp [hr] | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
| Mathlib.Analysis.Seminorm.724_0.ywwMCgoKeIFKDZ3 | @[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
xβ y : E
rβ : β
p : Seminorm π E
c : ββ₯0
hc : 0 < c
r : β
x : E
β’ ball (c β’ p) x r = ball p x (r / βc) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
| ext | theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
| Mathlib.Analysis.Seminorm.735_0.ywwMCgoKeIFKDZ3 | theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
xβΒΉ y : E
rβ : β
p : Seminorm π E
c : ββ₯0
hc : 0 < c
r : β
x xβ : E
β’ xβ β ball (c β’ p) x r β xβ β ball p x (r / βc) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
| rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)] | theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
| Mathlib.Analysis.Seminorm.735_0.ywwMCgoKeIFKDZ3 | theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
xβ y : E
rβ : β
p : Seminorm π E
c : ββ₯0
hc : 0 < c
r : β
x : E
β’ closedBall (c β’ p) x r = closedBall p x (r / βc) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.ball_smul Seminorm.ball_smul
theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
| ext | theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
| Mathlib.Analysis.Seminorm.742_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) | Mathlib_Analysis_Seminorm |
case h
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
xβΒΉ y : E
rβ : β
p : Seminorm π E
c : ββ₯0
hc : 0 < c
r : β
x xβ : E
β’ xβ β closedBall (c β’ p) x r β xβ β closedBall p x (r / βc) | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.ball_smul Seminorm.ball_smul
theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
ext
| rw [mem_closedBall, mem_closedBall, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
le_div_iff (NNReal.coe_pos.mpr hc)] | theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
ext
| Mathlib.Analysis.Seminorm.742_0.ywwMCgoKeIFKDZ3 | theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
x y : E
rβ : β
p q : Seminorm π E
e : E
r : β
β’ ball (p β q) e r = ball p e r β© ball q e r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.ball_smul Seminorm.ball_smul
theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
ext
rw [mem_closedBall, mem_closedBall, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
le_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.closed_ball_smul Seminorm.closedBall_smul
theorem ball_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
ball (p β q) e r = ball p e r β© ball q e r := by
| simp_rw [ball, β Set.setOf_and, coe_sup, Pi.sup_apply, sup_lt_iff] | theorem ball_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
ball (p β q) e r = ball p e r β© ball q e r := by
| Mathlib.Analysis.Seminorm.749_0.ywwMCgoKeIFKDZ3 | theorem ball_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
ball (p β q) e r = ball p e r β© ball q e r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
x y : E
rβ : β
p q : Seminorm π E
e : E
r : β
β’ closedBall (p β q) e r = closedBall p e r β© closedBall q e r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.ball_smul Seminorm.ball_smul
theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
ext
rw [mem_closedBall, mem_closedBall, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
le_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.closed_ball_smul Seminorm.closedBall_smul
theorem ball_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
ball (p β q) e r = ball p e r β© ball q e r := by
simp_rw [ball, β Set.setOf_and, coe_sup, Pi.sup_apply, sup_lt_iff]
#align seminorm.ball_sup Seminorm.ball_sup
theorem closedBall_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
closedBall (p β q) e r = closedBall p e r β© closedBall q e r := by
| simp_rw [closedBall, β Set.setOf_and, coe_sup, Pi.sup_apply, sup_le_iff] | theorem closedBall_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
closedBall (p β q) e r = closedBall p e r β© closedBall q e r := by
| Mathlib.Analysis.Seminorm.754_0.ywwMCgoKeIFKDZ3 | theorem closedBall_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
closedBall (p β q) e r = closedBall p e r β© closedBall q e r | Mathlib_Analysis_Seminorm |
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
x y : E
rβ : β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
H : Finset.Nonempty s
e : E
r : β
β’ ball (Finset.sup' s H p) e r = Finset.inf' s H fun i => ball (p i) e r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.ball_smul Seminorm.ball_smul
theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
ext
rw [mem_closedBall, mem_closedBall, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
le_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.closed_ball_smul Seminorm.closedBall_smul
theorem ball_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
ball (p β q) e r = ball p e r β© ball q e r := by
simp_rw [ball, β Set.setOf_and, coe_sup, Pi.sup_apply, sup_lt_iff]
#align seminorm.ball_sup Seminorm.ball_sup
theorem closedBall_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
closedBall (p β q) e r = closedBall p e r β© closedBall q e r := by
simp_rw [closedBall, β Set.setOf_and, coe_sup, Pi.sup_apply, sup_le_iff]
#align seminorm.closed_ball_sup Seminorm.closedBall_sup
theorem ball_finset_sup' (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (H : s.Nonempty) (e : E) (r : β) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
| induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih | theorem ball_finset_sup' (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (H : s.Nonempty) (e : E) (r : β) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
| Mathlib.Analysis.Seminorm.759_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup' (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (H : s.Nonempty) (e : E) (r : β) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r | Mathlib_Analysis_Seminorm |
case hβ
R : Type u_1
R' : Type u_2
π : Type u_3
πβ : Type u_4
πβ : Type u_5
π : Type u_6
E : Type u_7
Eβ : Type u_8
Eβ : Type u_9
F : Type u_10
G : Type u_11
ΞΉ : Type u_12
instβΒ² : SeminormedRing π
instβΒΉ : AddCommGroup E
instβ : SMul π E
pβ : Seminorm π E
x y : E
rβ : β
p : ΞΉ β Seminorm π E
s : Finset ΞΉ
e : E
r : β
a : ΞΉ
β’ ball (Finset.sup' {a} (_ : Finset.Nonempty {a}) p) e r =
Finset.inf' {a} (_ : Finset.Nonempty {a}) fun i => ball (p i) e r | /-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, YaΓ«l Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
/-!
# Seminorms
This file defines seminorms.
A seminorm is a function to the reals which is positive-semidefinite, absolutely homogeneous, and
subadditive. They are closely related to convex sets, and a topological vector space is locally
convex if and only if its topology is induced by a family of seminorms.
## Main declarations
For a module over a normed ring:
* `Seminorm`: A function to the reals that is positive-semidefinite, absolutely homogeneous, and
subadditive.
* `normSeminorm π E`: The norm on `E` as a seminorm.
## References
* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
## Tags
seminorm, locally convex, LCTVS
-/
set_option autoImplicit true
open NormedField Set Filter
open scoped BigOperators NNReal Pointwise Topology Uniformity
variable {R R' π πβ πβ π E Eβ Eβ F G ΞΉ : Type*}
/-- A seminorm on a module over a normed ring is a function to the reals that is positive
semidefinite, positive homogeneous, and subadditive. -/
structure Seminorm (π : Type*) (E : Type*) [SeminormedRing π] [AddGroup E] [SMul π E] extends
AddGroupSeminorm E where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
smul' : β (a : π) (x : E), toFun (a β’ x) = βaβ * toFun x
#align seminorm Seminorm
attribute [nolint docBlame] Seminorm.toAddGroupSeminorm
/-- `SeminormClass F π E` states that `F` is a type of seminorms on the `π`-module `E`.
You should extend this class when you extend `Seminorm`. -/
class SeminormClass (F : Type*) (π E : outParam <| Type*) [SeminormedRing π] [AddGroup E]
[SMul π E] extends AddGroupSeminormClass F E β where
/-- The seminorm of a scalar multiplication is the product of the absolute value of the scalar
and the original seminorm. -/
map_smul_eq_mul (f : F) (a : π) (x : E) : f (a β’ x) = βaβ * f x
#align seminorm_class SeminormClass
export SeminormClass (map_smul_eq_mul)
-- Porting note: dangerous instances no longer exist
-- attribute [nolint dangerousInstance] SeminormClass.toAddGroupSeminormClass
section Of
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing π`. -/
def Seminorm.of [SeminormedRing π] [AddCommGroup E] [Module π E] (f : E β β)
(add_le : β x y : E, f (x + y) β€ f x + f y) (smul : β (a : π) (x : E), f (a β’ x) = βaβ * f x) :
Seminorm π E where
toFun := f
map_zero' := by rw [β zero_smul π (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [β neg_one_smul π, smul, norm_neg, β smul, one_smul]
#align seminorm.of Seminorm.of
/-- Alternative constructor for a `Seminorm` over a normed field `π` that only assumes `f 0 = 0`
and an inequality for the scalar multiplication. -/
def Seminorm.ofSMulLE [NormedField π] [AddCommGroup E] [Module π E] (f : E β β) (map_zero : f 0 = 0)
(add_le : β x y, f (x + y) β€ f x + f y) (smul_le : β (r : π) (x), f (r β’ x) β€ βrβ * f x) :
Seminorm π E :=
Seminorm.of f add_le fun r x => by
refine' le_antisymm (smul_le r x) _
by_cases h : r = 0
Β· simp [h, map_zero]
rw [β mul_le_mul_left (inv_pos.mpr (norm_pos_iff.mpr h))]
rw [inv_mul_cancel_leftβ (norm_ne_zero_iff.mpr h)]
specialize smul_le rβ»ΒΉ (r β’ x)
rw [norm_inv] at smul_le
convert smul_le
simp [h]
#align seminorm.of_smul_le Seminorm.ofSMulLE
end Of
namespace Seminorm
section SeminormedRing
variable [SeminormedRing π]
section AddGroup
variable [AddGroup E]
section SMul
variable [SMul π E]
instance instSeminormClass : SeminormClass (Seminorm π E) π E where
coe f := f.toFun
coe_injective' f g h := by
rcases f with β¨β¨_β©β©
rcases g with β¨β¨_β©β©
congr
map_zero f := f.map_zero'
map_add_le_add f := f.add_le'
map_neg_eq_map f := f.neg'
map_smul_eq_mul f := f.smul'
#align seminorm.seminorm_class Seminorm.instSeminormClass
/-- Helper instance for when there's too many metavariables to apply `FunLike.hasCoeToFun`. -/
instance instCoeFun : CoeFun (Seminorm π E) fun _ => E β β :=
FunLike.hasCoeToFun
@[ext]
theorem ext {p q : Seminorm π E} (h : β x, (p : E β β) x = q x) : p = q :=
FunLike.ext p q h
#align seminorm.ext Seminorm.ext
instance instZero : Zero (Seminorm π E) :=
β¨{ AddGroupSeminorm.instZeroAddGroupSeminorm.zero with
smul' := fun _ _ => (mul_zero _).symm }β©
@[simp]
theorem coe_zero : β(0 : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_zero Seminorm.coe_zero
@[simp]
theorem zero_apply (x : E) : (0 : Seminorm π E) x = 0 :=
rfl
#align seminorm.zero_apply Seminorm.zero_apply
instance : Inhabited (Seminorm π E) :=
β¨0β©
variable (p : Seminorm π E) (c : π) (x y : E) (r : β)
/-- Any action on `β` which factors through `ββ₯0` applies to a seminorm. -/
instance instSMul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] : SMul R (Seminorm π E) where
smul r p :=
{ r β’ p.toAddGroupSeminorm with
toFun := fun x => r β’ p x
smul' := fun _ _ => by
simp only [β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def, smul_eq_mul]
rw [map_smul_eq_mul, mul_left_comm] }
instance [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] [SMul R' β] [SMul R' ββ₯0]
[IsScalarTower R' ββ₯0 β] [SMul R R'] [IsScalarTower R R' β] :
IsScalarTower R R' (Seminorm π E) where
smul_assoc r a p := ext fun x => smul_assoc r a (p x)
theorem coe_smul [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E) :
β(r β’ p) = r β’ βp :=
rfl
#align seminorm.coe_smul Seminorm.coe_smul
@[simp]
theorem smul_apply [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p : Seminorm π E)
(x : E) : (r β’ p) x = r β’ p x :=
rfl
#align seminorm.smul_apply Seminorm.smul_apply
instance instAdd : Add (Seminorm π E) where
add p q :=
{ p.toAddGroupSeminorm + q.toAddGroupSeminorm with
toFun := fun x => p x + q x
smul' := fun a x => by simp only [map_smul_eq_mul, map_smul_eq_mul, mul_add] }
theorem coe_add (p q : Seminorm π E) : β(p + q) = p + q :=
rfl
#align seminorm.coe_add Seminorm.coe_add
@[simp]
theorem add_apply (p q : Seminorm π E) (x : E) : (p + q) x = p x + q x :=
rfl
#align seminorm.add_apply Seminorm.add_apply
instance instAddMonoid : AddMonoid (Seminorm π E) :=
FunLike.coe_injective.addMonoid _ rfl coe_add fun _ _ => by rfl
instance instOrderedCancelAddCommMonoid : OrderedCancelAddCommMonoid (Seminorm π E) :=
FunLike.coe_injective.orderedCancelAddCommMonoid _ rfl coe_add fun _ _ => rfl
instance instMulAction [Monoid R] [MulAction R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
MulAction R (Seminorm π E) :=
FunLike.coe_injective.mulAction _ (by intros; rfl)
variable (π E)
/-- `coeFn` as an `AddMonoidHom`. Helper definition for showing that `Seminorm π E` is a module. -/
@[simps]
def coeFnAddMonoidHom : AddMonoidHom (Seminorm π E) (E β β) where
toFun := (β)
map_zero' := coe_zero
map_add' := coe_add
#align seminorm.coe_fn_add_monoid_hom Seminorm.coeFnAddMonoidHom
theorem coeFnAddMonoidHom_injective : Function.Injective (coeFnAddMonoidHom π E) :=
show @Function.Injective (Seminorm π E) (E β β) (β) from FunLike.coe_injective
#align seminorm.coe_fn_add_monoid_hom_injective Seminorm.coeFnAddMonoidHom_injective
variable {π E}
instance instDistribMulAction [Monoid R] [DistribMulAction R β] [SMul R ββ₯0]
[IsScalarTower R ββ₯0 β] : DistribMulAction R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).distribMulAction _ (by intros; rfl)
instance instModule [Semiring R] [Module R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] :
Module R (Seminorm π E) :=
(coeFnAddMonoidHom_injective π E).module R _ (by intros; rfl)
instance instSup : Sup (Seminorm π E) where
sup p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := p β q
smul' := fun x v =>
(congr_argβ max (map_smul_eq_mul p x v) (map_smul_eq_mul q x v)).trans <|
(mul_max_of_nonneg _ _ <| norm_nonneg x).symm }
@[simp]
theorem coe_sup (p q : Seminorm π E) : β(p β q) = (p : E β β) β (q : E β β) :=
rfl
#align seminorm.coe_sup Seminorm.coe_sup
theorem sup_apply (p q : Seminorm π E) (x : E) : (p β q) x = p x β q x :=
rfl
#align seminorm.sup_apply Seminorm.sup_apply
theorem smul_sup [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q :=
have real.smul_max : β x y : β, r β’ max x y = max (r β’ x) (r β’ y) := fun x y => by
simpa only [β smul_eq_mul, β NNReal.smul_def, smul_one_smul ββ₯0 r (_ : β)] using
mul_max_of_nonneg x y (r β’ (1 : ββ₯0) : ββ₯0).coe_nonneg
ext fun x => real.smul_max _ _
#align seminorm.smul_sup Seminorm.smul_sup
instance instPartialOrder : PartialOrder (Seminorm π E) :=
PartialOrder.lift _ FunLike.coe_injective
@[simp, norm_cast]
theorem coe_le_coe {p q : Seminorm π E} : (p : E β β) β€ q β p β€ q :=
Iff.rfl
#align seminorm.coe_le_coe Seminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_lt_coe {p q : Seminorm π E} : (p : E β β) < q β p < q :=
Iff.rfl
#align seminorm.coe_lt_coe Seminorm.coe_lt_coe
theorem le_def {p q : Seminorm π E} : p β€ q β β x, p x β€ q x :=
Iff.rfl
#align seminorm.le_def Seminorm.le_def
theorem lt_def {p q : Seminorm π E} : p < q β p β€ q β§ β x, p x < q x :=
@Pi.lt_def _ _ _ p q
#align seminorm.lt_def Seminorm.lt_def
instance instSemilatticeSup : SemilatticeSup (Seminorm π E) :=
Function.Injective.semilatticeSup _ FunLike.coe_injective coe_sup
end SMul
end AddGroup
section Module
variable [SeminormedRing πβ] [SeminormedRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : πβ β+* πβ} [RingHomIsometric Οββ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [AddCommGroup Eβ]
variable [AddCommGroup F] [AddCommGroup G]
variable [Module π E] [Module πβ Eβ] [Module πβ Eβ] [Module π F] [Module π G]
-- Porting note: even though this instance is found immediately by typeclass search,
-- it seems to be needed below!?
noncomputable instance smul_nnreal_real : SMul ββ₯0 β := inferInstance
variable [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β]
/-- Composition of a seminorm with a linear map is a seminorm. -/
def comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : Seminorm π E :=
{ p.toAddGroupSeminorm.comp f.toAddMonoidHom with
toFun := fun x => p (f x)
-- Porting note: the `simp only` below used to be part of the `rw`.
-- I'm not sure why this change was needed, and am worried by it!
smul' := fun _ _ => by simp only [map_smulββ]; rw [map_smul_eq_mul, RingHomIsometric.is_iso] }
#align seminorm.comp Seminorm.comp
theorem coe_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) : β(p.comp f) = p β f :=
rfl
#align seminorm.coe_comp Seminorm.coe_comp
@[simp]
theorem comp_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (x : E) : (p.comp f) x = p (f x) :=
rfl
#align seminorm.comp_apply Seminorm.comp_apply
@[simp]
theorem comp_id (p : Seminorm π E) : p.comp LinearMap.id = p :=
ext fun _ => rfl
#align seminorm.comp_id Seminorm.comp_id
@[simp]
theorem comp_zero (p : Seminorm πβ Eβ) : p.comp (0 : E βββ[Οββ] Eβ) = 0 :=
ext fun _ => map_zero p
#align seminorm.comp_zero Seminorm.comp_zero
@[simp]
theorem zero_comp (f : E βββ[Οββ] Eβ) : (0 : Seminorm πβ Eβ).comp f = 0 :=
ext fun _ => rfl
#align seminorm.zero_comp Seminorm.zero_comp
theorem comp_comp [RingHomCompTriple Οββ Οββ Οββ] (p : Seminorm πβ Eβ) (g : Eβ βββ[Οββ] Eβ)
(f : E βββ[Οββ] Eβ) : p.comp (g.comp f) = (p.comp g).comp f :=
ext fun _ => rfl
#align seminorm.comp_comp Seminorm.comp_comp
theorem add_comp (p q : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) :
(p + q).comp f = p.comp f + q.comp f :=
ext fun _ => rfl
#align seminorm.add_comp Seminorm.add_comp
theorem comp_add_le (p : Seminorm πβ Eβ) (f g : E βββ[Οββ] Eβ) :
p.comp (f + g) β€ p.comp f + p.comp g := fun _ => map_add_le_add p _ _
#align seminorm.comp_add_le Seminorm.comp_add_le
theorem smul_comp (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : R) :
(c β’ p).comp f = c β’ p.comp f :=
ext fun _ => rfl
#align seminorm.smul_comp Seminorm.smul_comp
theorem comp_mono {p q : Seminorm πβ Eβ} (f : E βββ[Οββ] Eβ) (hp : p β€ q) : p.comp f β€ q.comp f :=
fun _ => hp _
#align seminorm.comp_mono Seminorm.comp_mono
/-- The composition as an `AddMonoidHom`. -/
@[simps]
def pullback (f : E βββ[Οββ] Eβ) : Seminorm πβ Eβ β+ Seminorm π E where
toFun := fun p => p.comp f
map_zero' := zero_comp f
map_add' := fun p q => add_comp p q f
#align seminorm.pullback Seminorm.pullback
instance instOrderBot : OrderBot (Seminorm π E) where
bot := 0
bot_le := map_nonneg
@[simp]
theorem coe_bot : β(β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.coe_bot Seminorm.coe_bot
theorem bot_eq_zero : (β₯ : Seminorm π E) = 0 :=
rfl
#align seminorm.bot_eq_zero Seminorm.bot_eq_zero
theorem smul_le_smul {p q : Seminorm π E} {a b : ββ₯0} (hpq : p β€ q) (hab : a β€ b) :
a β’ p β€ b β’ q := by
simp_rw [le_def]
intro x
exact mul_le_mul hab (hpq x) (map_nonneg p x) (NNReal.coe_nonneg b)
#align seminorm.smul_le_smul Seminorm.smul_le_smul
theorem finset_sup_apply (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = β(s.sup fun i => β¨p i x, map_nonneg (p i) xβ© : ββ₯0) := by
induction' s using Finset.cons_induction_on with a s ha ih
Β· rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply]
norm_cast
Β· rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,
NNReal.coe_max, NNReal.coe_mk, ih]
#align seminorm.finset_sup_apply Seminorm.finset_sup_apply
theorem exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) {s : Finset ΞΉ} (hs : s.Nonempty) (x : E) :
β i β s, s.sup p x = p i x := by
rcases Finset.exists_mem_eq_sup s hs (fun i β¦ (β¨p i x, map_nonneg _ _β© : ββ₯0)) with β¨i, hi, hixβ©
rw [finset_sup_apply]
exact β¨i, hi, congr_arg _ hixβ©
theorem zero_or_exists_apply_eq_finset_sup (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (x : E) :
s.sup p x = 0 β¨ β i β s, s.sup p x = p i x := by
rcases Finset.eq_empty_or_nonempty s with (rfl|hs)
Β· left; rfl
Β· right; exact exists_apply_eq_finset_sup p hs x
theorem finset_sup_smul (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (C : ββ₯0) :
s.sup (C β’ p) = C β’ s.sup p := by
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((β) : ββ₯0 β β) (NNReal.mul_finset_sup C s (fun i β¦ β¨p i x, map_nonneg _ _β©))
theorem finset_sup_le_sum (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) : s.sup p β€ β i in s, p i := by
classical
refine' Finset.sup_le_iff.mpr _
intro i hi
rw [Finset.sum_eq_sum_diff_singleton_add hi, le_add_iff_nonneg_left]
exact bot_le
#align seminorm.finset_sup_le_sum Seminorm.finset_sup_le_sum
theorem finset_sup_apply_le {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 β€ a)
(h : β i, i β s β p i x β€ a) : s.sup p x β€ a := by
lift a to ββ₯0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h
#align seminorm.finset_sup_apply_le Seminorm.finset_sup_apply_le
theorem le_finset_sup_apply {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {i : ΞΉ}
(hi : i β s) : p i x β€ s.sup p x :=
(Finset.le_sup hi : p i β€ s.sup p) x
theorem finset_sup_apply_lt {p : ΞΉ β Seminorm π E} {s : Finset ΞΉ} {x : E} {a : β} (ha : 0 < a)
(h : β i, i β s β p i x < a) : s.sup p x < a := by
lift a to ββ₯0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
Β· exact h
Β· exact NNReal.coe_pos.mpr ha
#align seminorm.finset_sup_apply_lt Seminorm.finset_sup_apply_lt
theorem norm_sub_map_le_sub (p : Seminorm π E) (x y : E) : βp x - p yβ β€ p (x - y) :=
abs_sub_map_le_sub p x y
#align seminorm.norm_sub_map_le_sub Seminorm.norm_sub_map_le_sub
end Module
end SeminormedRing
section SeminormedCommRing
variable [SeminormedRing π] [SeminormedCommRing πβ]
variable {Οββ : π β+* πβ} [RingHomIsometric Οββ]
variable [AddCommGroup E] [AddCommGroup Eβ] [Module π E] [Module πβ Eβ]
theorem comp_smul (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) :
p.comp (c β’ f) = βcββ β’ p.comp f :=
ext fun _ => by
rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,
smul_eq_mul, comp_apply]
#align seminorm.comp_smul Seminorm.comp_smul
theorem comp_smul_apply (p : Seminorm πβ Eβ) (f : E βββ[Οββ] Eβ) (c : πβ) (x : E) :
p.comp (c β’ f) x = βcβ * p (f x) :=
map_smul_eq_mul p _ _
#align seminorm.comp_smul_apply Seminorm.comp_smul_apply
end SeminormedCommRing
section NormedField
variable [NormedField π] [AddCommGroup E] [Module π E] {p q : Seminorm π E} {x : E}
/-- Auxiliary lemma to show that the infimum of seminorms is well-defined. -/
theorem bddBelow_range_add : BddBelow (range fun u => p u + q (x - u)) :=
β¨0, by
rintro _ β¨x, rflβ©
dsimp; positivityβ©
#align seminorm.bdd_below_range_add Seminorm.bddBelow_range_add
noncomputable instance instInf : Inf (Seminorm π E) where
inf p q :=
{ p.toAddGroupSeminorm β q.toAddGroupSeminorm with
toFun := fun x => β¨
u : E, p u + q (x - u)
smul' := by
intro a x
obtain rfl | ha := eq_or_ne a 0
Β· rw [norm_zero, zero_mul, zero_smul]
refine'
ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
-- Porting note: the following was previously `fun i => by positivity`
(fun i => add_nonneg (map_nonneg _ _) (map_nonneg _ _))
fun x hx => β¨0, by rwa [map_zero, sub_zero, map_zero, add_zero]β©
simp_rw [Real.mul_iInf_of_nonneg (norm_nonneg a), mul_add, β map_smul_eq_mul p, β
map_smul_eq_mul q, smul_sub]
refine'
Function.Surjective.iInf_congr ((aβ»ΒΉ β’ Β·) : E β E)
(fun u => β¨a β’ u, inv_smul_smulβ ha uβ©) fun u => _
rw [smul_inv_smulβ ha] }
@[simp]
theorem inf_apply (p q : Seminorm π E) (x : E) : (p β q) x = β¨
u : E, p u + q (x - u) :=
rfl
#align seminorm.inf_apply Seminorm.inf_apply
noncomputable instance instLattice : Lattice (Seminorm π E) :=
{ Seminorm.instSemilatticeSup with
inf := (Β· β Β·)
inf_le_left := fun p q x =>
ciInf_le_of_le bddBelow_range_add x <| by
simp only [sub_self, map_zero, add_zero]; rfl
inf_le_right := fun p q x =>
ciInf_le_of_le bddBelow_range_add 0 <| by
simp only [sub_self, map_zero, zero_add, sub_zero]; rfl
le_inf := fun a b c hab hac x =>
le_ciInf fun u => (le_map_add_map_sub a _ _).trans <| add_le_add (hab _) (hac _) }
theorem smul_inf [SMul R β] [SMul R ββ₯0] [IsScalarTower R ββ₯0 β] (r : R) (p q : Seminorm π E) :
r β’ (p β q) = r β’ p β r β’ q := by
ext
simp_rw [smul_apply, inf_apply, smul_apply, β smul_one_smul ββ₯0 r (_ : β), NNReal.smul_def,
smul_eq_mul, Real.mul_iInf_of_nonneg (NNReal.coe_nonneg _), mul_add]
#align seminorm.smul_inf Seminorm.smul_inf
section Classical
open Classical
/-- We define the supremum of an arbitrary subset of `Seminorm π E` as follows:
* if `s` is `BddAbove` *as a set of functions `E β β`* (that is, if `s` is pointwise bounded
above), we take the pointwise supremum of all elements of `s`, and we prove that it is indeed a
seminorm.
* otherwise, we take the zero seminorm `β₯`.
There are two things worth mentioning here:
* First, it is not trivial at first that `s` being bounded above *by a function* implies
being bounded above *as a seminorm*. We show this in `Seminorm.bddAbove_iff` by using
that the `Sup s` as defined here is then a bounding seminorm for `s`. So it is important to make
the case disjunction on `BddAbove ((β) '' s : Set (E β β))` and not `BddAbove s`.
* Since the pointwise `Sup` already gives `0` at points where a family of functions is
not bounded above, one could hope that just using the pointwise `Sup` would work here, without the
need for an additional case disjunction. As discussed on Zulip, this doesn't work because this can
give a function which does *not* satisfy the seminorm axioms (typically sub-additivity).
-/
noncomputable instance instSupSet : SupSet (Seminorm π E) where
sSup s :=
if h : BddAbove ((β) '' s : Set (E β β)) then
{ toFun := β¨ p : s, ((p : Seminorm π E) : E β β)
map_zero' := by
rw [iSup_apply, β @Real.ciSup_const_zero s]
congr!
rename_i _ _ _ i
exact map_zero i.1
add_le' := fun x y => by
rcases h with β¨q, hqβ©
obtain rfl | h := s.eq_empty_or_nonempty
Β· simp [Real.ciSup_empty]
haveI : Nonempty βs := h.coe_sort
simp only [iSup_apply]
refine' ciSup_le fun i =>
((i : Seminorm π E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun x) β¨q x, _β© i)
(le_ciSup (f := fun i => (Subtype.val i : Seminorm π E).toFun y) β¨q y, _β© i)
<;> rw [mem_upperBounds, forall_range_iff]
<;> exact fun j => hq (mem_image_of_mem _ j.2) _
neg' := fun x => by
simp only [iSup_apply]
congr! 2
rename_i _ _ _ i
exact i.1.neg' _
smul' := fun a x => by
simp only [iSup_apply]
rw [β smul_eq_mul,
Real.smul_iSup_of_nonneg (norm_nonneg a) fun i : s => (i : Seminorm π E) x]
congr!
rename_i _ _ _ i
exact i.1.smul' a x }
else β₯
protected theorem coe_sSup_eq' {s : Set <| Seminorm π E}
(hs : BddAbove ((β) '' s : Set (E β β))) : β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
congr_arg _ (dif_pos hs)
#align seminorm.coe_Sup_eq' Seminorm.coe_sSup_eq'
protected theorem bddAbove_iff {s : Set <| Seminorm π E} :
BddAbove s β BddAbove ((β) '' s : Set (E β β)) :=
β¨fun β¨q, hqβ© => β¨q, ball_image_of_ball fun p hp => hq hpβ©, fun H =>
β¨sSup s, fun p hp x => by
dsimp
rw [Seminorm.coe_sSup_eq' H, iSup_apply]
rcases H with β¨q, hqβ©
exact
le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) xβ© β¨p, hpβ©β©β©
#align seminorm.bdd_above_iff Seminorm.bddAbove_iff
protected theorem bddAbove_range_iff {p : ΞΉ β Seminorm π E} :
BddAbove (range p) β β x, BddAbove (range fun i β¦ p i x) := by
rw [Seminorm.bddAbove_iff, β range_comp, bddAbove_range_pi]; rfl
protected theorem coe_sSup_eq {s : Set <| Seminorm π E} (hs : BddAbove s) :
β(sSup s) = β¨ p : s, ((p : Seminorm π E) : E β β) :=
Seminorm.coe_sSup_eq' (Seminorm.bddAbove_iff.mp hs)
#align seminorm.coe_Sup_eq Seminorm.coe_sSup_eq
protected theorem coe_iSup_eq {ΞΉ : Type*} {p : ΞΉ β Seminorm π E} (hp : BddAbove (range p)) :
β(β¨ i, p i) = β¨ i, ((p i : Seminorm π E) : E β β) := by
rw [β sSup_range, Seminorm.coe_sSup_eq hp]
exact iSup_range' (fun p : Seminorm π E => (p : E β β)) p
#align seminorm.coe_supr_eq Seminorm.coe_iSup_eq
protected theorem sSup_apply {s : Set (Seminorm π E)} (hp : BddAbove s) {x : E} :
(sSup s) x = β¨ p : s, (p : E β β) x := by
rw [Seminorm.coe_sSup_eq hp, iSup_apply]
protected theorem iSup_apply {ΞΉ : Type*} {p : ΞΉ β Seminorm π E}
(hp : BddAbove (range p)) {x : E} : (β¨ i, p i) x = β¨ i, p i x := by
rw [Seminorm.coe_iSup_eq hp, iSup_apply]
protected theorem sSup_empty : sSup (β
: Set (Seminorm π E)) = β₯ := by
ext
rw [Seminorm.sSup_apply bddAbove_empty, Real.ciSup_empty]
rfl
private theorem Seminorm.isLUB_sSup (s : Set (Seminorm π E)) (hsβ : BddAbove s) (hsβ : s.Nonempty) :
IsLUB s (sSup s) := by
refine' β¨fun p hp x => _, fun p hp x => _β© <;> haveI : Nonempty βs := hsβ.coe_sort <;>
dsimp <;> rw [Seminorm.coe_sSup_eq hsβ, iSup_apply]
Β· rcases hsβ with β¨q, hqβ©
exact le_ciSup β¨q x, forall_range_iff.mpr fun i : s => hq i.2 xβ© β¨p, hpβ©
Β· exact ciSup_le fun q => hp q.2 x
/-- `Seminorm π E` is a conditionally complete lattice.
Note that, while `inf`, `sup` and `sSup` have good definitional properties (corresponding to
the instances given here for `Inf`, `Sup` and `SupSet` respectively), `sInf s` is just
defined as the supremum of the lower bounds of `s`, which is not really useful in practice. If you
need to use `sInf` on seminorms, then you should probably provide a more workable definition first,
but this is unlikely to happen so we keep the "bad" definition for now. -/
noncomputable instance instConditionallyCompleteLattice :
ConditionallyCompleteLattice (Seminorm π E) :=
conditionallyCompleteLatticeOfLatticeOfsSup (Seminorm π E) Seminorm.isLUB_sSup
end Classical
end NormedField
/-! ### Seminorm ball -/
section SeminormedRing
variable [SeminormedRing π]
section AddCommGroup
variable [AddCommGroup E]
section SMul
variable [SMul π E] (p : Seminorm π E)
/-- The ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y` with
`p (y - x) < r`. -/
def ball (x : E) (r : β) :=
{ y : E | p (y - x) < r }
#align seminorm.ball Seminorm.ball
/-- The closed ball of radius `r` at `x` with respect to seminorm `p` is the set of elements `y`
with `p (y - x) β€ r`. -/
def closedBall (x : E) (r : β) :=
{ y : E | p (y - x) β€ r }
#align seminorm.closed_ball Seminorm.closedBall
variable {x y : E} {r : β}
@[simp]
theorem mem_ball : y β ball p x r β p (y - x) < r :=
Iff.rfl
#align seminorm.mem_ball Seminorm.mem_ball
@[simp]
theorem mem_closedBall : y β closedBall p x r β p (y - x) β€ r :=
Iff.rfl
#align seminorm.mem_closed_ball Seminorm.mem_closedBall
theorem mem_ball_self (hr : 0 < r) : x β ball p x r := by simp [hr]
#align seminorm.mem_ball_self Seminorm.mem_ball_self
theorem mem_closedBall_self (hr : 0 β€ r) : x β closedBall p x r := by simp [hr]
#align seminorm.mem_closed_ball_self Seminorm.mem_closedBall_self
theorem mem_ball_zero : y β ball p 0 r β p y < r := by rw [mem_ball, sub_zero]
#align seminorm.mem_ball_zero Seminorm.mem_ball_zero
theorem mem_closedBall_zero : y β closedBall p 0 r β p y β€ r := by rw [mem_closedBall, sub_zero]
#align seminorm.mem_closed_ball_zero Seminorm.mem_closedBall_zero
theorem ball_zero_eq : ball p 0 r = { y : E | p y < r } :=
Set.ext fun _ => p.mem_ball_zero
#align seminorm.ball_zero_eq Seminorm.ball_zero_eq
theorem closedBall_zero_eq : closedBall p 0 r = { y : E | p y β€ r } :=
Set.ext fun _ => p.mem_closedBall_zero
#align seminorm.closed_ball_zero_eq Seminorm.closedBall_zero_eq
theorem ball_subset_closedBall (x r) : ball p x r β closedBall p x r := fun _ h =>
(mem_closedBall _).mpr ((mem_ball _).mp h).le
#align seminorm.ball_subset_closed_ball Seminorm.ball_subset_closedBall
theorem closedBall_eq_biInter_ball (x r) : closedBall p x r = β Ο > r, ball p x Ο := by
ext y; simp_rw [mem_closedBall, mem_iInterβ, mem_ball, β forall_lt_iff_le']
#align seminorm.closed_ball_eq_bInter_ball Seminorm.closedBall_eq_biInter_ball
@[simp]
theorem ball_zero' (x : E) (hr : 0 < r) : ball (0 : Seminorm π E) x r = Set.univ := by
rw [Set.eq_univ_iff_forall, ball]
simp [hr]
#align seminorm.ball_zero' Seminorm.ball_zero'
@[simp]
theorem closedBall_zero' (x : E) (hr : 0 < r) : closedBall (0 : Seminorm π E) x r = Set.univ :=
eq_univ_of_subset (ball_subset_closedBall _ _ _) (ball_zero' x hr)
#align seminorm.closed_ball_zero' Seminorm.closedBall_zero'
theorem ball_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).ball x r = p.ball x (r / c) := by
ext
rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
lt_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.ball_smul Seminorm.ball_smul
theorem closedBall_smul (p : Seminorm π E) {c : NNReal} (hc : 0 < c) (r : β) (x : E) :
(c β’ p).closedBall x r = p.closedBall x (r / c) := by
ext
rw [mem_closedBall, mem_closedBall, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
le_div_iff (NNReal.coe_pos.mpr hc)]
#align seminorm.closed_ball_smul Seminorm.closedBall_smul
theorem ball_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
ball (p β q) e r = ball p e r β© ball q e r := by
simp_rw [ball, β Set.setOf_and, coe_sup, Pi.sup_apply, sup_lt_iff]
#align seminorm.ball_sup Seminorm.ball_sup
theorem closedBall_sup (p : Seminorm π E) (q : Seminorm π E) (e : E) (r : β) :
closedBall (p β q) e r = closedBall p e r β© closedBall q e r := by
simp_rw [closedBall, β Set.setOf_and, coe_sup, Pi.sup_apply, sup_le_iff]
#align seminorm.closed_ball_sup Seminorm.closedBall_sup
theorem ball_finset_sup' (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (H : s.Nonempty) (e : E) (r : β) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
Β· | classical simp | theorem ball_finset_sup' (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (H : s.Nonempty) (e : E) (r : β) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r := by
induction' H using Finset.Nonempty.cons_induction with a a s ha hs ih
Β· | Mathlib.Analysis.Seminorm.759_0.ywwMCgoKeIFKDZ3 | theorem ball_finset_sup' (p : ΞΉ β Seminorm π E) (s : Finset ΞΉ) (H : s.Nonempty) (e : E) (r : β) :
ball (s.sup' H p) e r = s.inf' H fun i => ball (p i) e r | Mathlib_Analysis_Seminorm |
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