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Mathlib.Topology.Algebra.Module.WeakBilin

Weak dual topology #

This file defines the weak topology given two vector spaces E and F over a commutative semiring 𝕜 and a bilinear form B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜. The weak topology on E is the coarsest topology such that for all y : F every map fun x => B x y is continuous.

Main definitions #

The main definition is the type WeakBilin B.

Main results #

We establish that WeakBilin B has the following structure:

We prove the following results characterizing the weak topology:

Notations #

No new notation is introduced.

References #

Tags #

weak-star, weak dual, duality

instance WeakBilin.instAddCommMonoid {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [a : AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
Equations
instance WeakBilin.instModule {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [AddCommMonoid E] [m : Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
Module 𝕜 (WeakBilin B)
Equations
instance WeakBilin.instAddCommGroup {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [a : AddCommGroup E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
Equations
@[instance 100]
instance WeakBilin.instModule' {𝕜 : Type u_2} {𝕝 : Type u_3} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [m : Module 𝕝 E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
Module 𝕝 (WeakBilin B)
Equations
instance WeakBilin.instIsScalarTower {𝕜 : Type u_2} {𝕝 : Type u_3} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [SMul 𝕝 𝕜] [Module 𝕝 E] [s : IsScalarTower 𝕝 𝕜 E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
IsScalarTower 𝕝 𝕜 (WeakBilin B)
instance WeakBilin.instTopologicalSpace {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
Equations
theorem WeakBilin.coeFn_continuous {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
Continuous fun (x : WeakBilin B) (y : F) => (B x) y

The coercion (fun x y => B x y) : E → (F → 𝕜) is continuous.

theorem WeakBilin.eval_continuous {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (y : F) :
Continuous fun (x : WeakBilin B) => (B x) y
theorem WeakBilin.continuous_of_continuous_eval {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) [TopologicalSpace α] {g : αWeakBilin B} (h : ∀ (y : F), Continuous fun (a : α) => (B (g a)) y) :
theorem WeakBilin.isEmbedding {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective B) :
Topology.IsEmbedding fun (x : WeakBilin B) (y : F) => (B x) y

The coercion (fun x y => B x y) : E → (F → 𝕜) is an embedding.

@[deprecated WeakBilin.isEmbedding (since := "2024-10-26")]
theorem WeakBilin.embedding {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective B) :
Topology.IsEmbedding fun (x : WeakBilin B) (y : F) => (B x) y

Alias of WeakBilin.isEmbedding.


The coercion (fun x y => B x y) : E → (F → 𝕜) is an embedding.

theorem WeakBilin.tendsto_iff_forall_eval_tendsto {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) {l : Filter α} {f : αWeakBilin B} {x : WeakBilin B} (hB : Function.Injective B) :
Filter.Tendsto f l (nhds x) ∀ (y : F), Filter.Tendsto (fun (i : α) => (B (f i)) y) l (nhds ((B x) y))
instance WeakBilin.instContinuousAdd {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) [ContinuousAdd 𝕜] :

Addition in WeakBilin B is continuous.

instance WeakBilin.instContinuousSMul {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) [ContinuousSMul 𝕜 𝕜] :

Scalar multiplication by 𝕜 on WeakBilin B is continuous.

def WeakBilin.eval {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] :
F →ₗ[𝕜] WeakBilin B →L[𝕜] 𝕜

Map F into the topological dual of E with the weak topology induced by F

Equations
  • WeakBilin.eval B = { toFun := fun (f : F) => { toLinearMap := B.flip f, cont := }, map_add' := , map_smul' := }
instance WeakBilin.instIsTopologicalAddGroup {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [TopologicalSpace 𝕜] [CommRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) [ContinuousAdd 𝕜] :

WeakBilin B is a IsTopologicalAddGroup, meaning that addition and negation are continuous.