Weak dual topology

We continue in the setting of Mathlib/Topology/Algebra/Module/WeakBilin.lean, which 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.

In this file, we consider two special cases. In the case that F = E →L[𝕜] 𝕜 and B being the canonical pairing, we obtain the weak-* topology, WeakDual 𝕜 E := (E →L[𝕜] 𝕜). Interchanging the arguments in the bilinear form yields the weak topology WeakSpace 𝕜 E := E.

Main definitions

The main definitions are the types WeakDual 𝕜 E and WeakSpace 𝕜 E, with the respective topology instances on it.

• WeakDual 𝕜 E is a type synonym for Dual 𝕜 E (when the latter is defined): both are equal to the type E →L[𝕜] 𝕜 of continuous linear maps from a module E over 𝕜 to the ring 𝕜.
• The instance WeakDual.instTopologicalSpace is the weak-* topology on WeakDual 𝕜 E, i.e., the coarsest topology making the evaluation maps at all z : E continuous.
• WeakSpace 𝕜 E is a type synonym for E (when the latter is defined).
• The instance WeakSpace.instTopologicalSpace is the weak topology on E, i.e., the coarsest topology such that all v : dual 𝕜 E remain continuous.

Notations

No new notation is introduced.

References

• [H. H. Schaefer, Topological Vector Spaces][schaefer1966]

Tags

weak-star, weak dual, duality

def topDualPairing (𝕜 : Type u_6) (E : Type u_7) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 𝕜] : (E →L[𝕜] 𝕜) →ₗ[𝕜] E →ₗ[𝕜] 𝕜

The canonical pairing of a vector space and its topological dual.

Equations

• topDualPairing 𝕜 E = ContinuousLinearMap.coeLM 𝕜

theorem topDualPairing_apply {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 𝕜] (v : E →L[𝕜] 𝕜) (x : E) : ((topDualPairing 𝕜 E) v) x = v x

def WeakDual (𝕜 : Type u_6) (E : Type u_7) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Type (max u_6 u_7)

The weak star topology is the topology coarsest topology on E →L[𝕜] 𝕜 such that all functionals fun v => v x are continuous.

Equations

• WeakDual 𝕜 E = WeakBilin (topDualPairing 𝕜 E)

instance WeakDual.instAddCommMonoid {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : AddCommMonoid (WeakDual 𝕜 E)

Equations

• WeakDual.instAddCommMonoid = WeakBilin.instAddCommMonoid (topDualPairing 𝕜 E)

instance WeakDual.instModule {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Module 𝕜 (WeakDual 𝕜 E)

Equations

• WeakDual.instModule = WeakBilin.instModule (topDualPairing 𝕜 E)

instance WeakDual.instTopologicalSpace {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : TopologicalSpace (WeakDual 𝕜 E)

Equations

• WeakDual.instTopologicalSpace = WeakBilin.instTopologicalSpace (topDualPairing 𝕜 E)

instance WeakDual.instContinuousAdd {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : ContinuousAdd (WeakDual 𝕜 E)

instance WeakDual.instInhabited {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Inhabited (WeakDual 𝕜 E)

Equations

• WeakDual.instInhabited = ContinuousLinearMap.inhabited

instance WeakDual.instFunLike {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : FunLike (WeakDual 𝕜 E) E 𝕜

Equations

• WeakDual.instFunLike = ContinuousLinearMap.funLike

instance WeakDual.instContinuousLinearMapClass {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : ContinuousLinearMapClass (WeakDual 𝕜 E) 𝕜 E 𝕜

instance WeakDual.instMulAction {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (M : Type u_6) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜] [ContinuousConstSMul M 𝕜] : MulAction M (WeakDual 𝕜 E)

If a monoid M distributively continuously acts on 𝕜 and this action commutes with multiplication on 𝕜, then it acts on WeakDual 𝕜 E.

Equations

• WeakDual.instMulAction M = ContinuousLinearMap.mulAction

instance WeakDual.instDistribMulAction {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (M : Type u_6) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜] [ContinuousConstSMul M 𝕜] : DistribMulAction M (WeakDual 𝕜 E)

If a monoid M distributively continuously acts on 𝕜 and this action commutes with multiplication on 𝕜, then it acts distributively on WeakDual 𝕜 E.

Equations

• WeakDual.instDistribMulAction M = ContinuousLinearMap.distribMulAction

instance WeakDual.instModule' {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (R : Type u_6) [Semiring R] [Module R 𝕜] [SMulCommClass 𝕜 R 𝕜] [ContinuousConstSMul R 𝕜] : Module R (WeakDual 𝕜 E)

If 𝕜 is a topological module over a semiring R and scalar multiplication commutes with the multiplication on 𝕜, then WeakDual 𝕜 E is a module over R.

Equations

• WeakDual.instModule' R = ContinuousLinearMap.module

instance WeakDual.instContinuousConstSMul {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (M : Type u_6) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜] [ContinuousConstSMul M 𝕜] : ContinuousConstSMul M (WeakDual 𝕜 E)

instance WeakDual.instContinuousSMul {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (M : Type u_6) [Monoid M] [DistribMulAction M 𝕜] [SMulCommClass 𝕜 M 𝕜] [TopologicalSpace M] [ContinuousSMul M 𝕜] : ContinuousSMul M (WeakDual 𝕜 E)

If a monoid M distributively continuously acts on 𝕜 and this action commutes with multiplication on 𝕜, then it continuously acts on WeakDual 𝕜 E.

theorem WeakDual.coeFn_continuous {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Continuous fun (x : WeakDual 𝕜 E) (y : E) => x y

theorem WeakDual.eval_continuous {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (y : E) : Continuous fun (x : WeakDual 𝕜 E) => x y

theorem WeakDual.continuous_of_continuous_eval {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalSpace α] {g : α → WeakDual 𝕜 E} (h : ∀ (y : E), Continuous fun (a : α) => (g a) y) : Continuous g

instance WeakDual.instT2Space {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [T2Space 𝕜] : T2Space (WeakDual 𝕜 E)

instance WeakDual.instAddCommGroup {𝕜 : Type u_2} {E : Type u_4} [CommRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalAddGroup 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] : AddCommGroup (WeakDual 𝕜 E)

Equations

• WeakDual.instAddCommGroup = WeakBilin.instAddCommGroup (topDualPairing 𝕜 E)

instance WeakDual.instIsTopologicalAddGroup {𝕜 : Type u_2} {E : Type u_4} [CommRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalAddGroup 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] : IsTopologicalAddGroup (WeakDual 𝕜 E)

def WeakSpace (𝕜 : Type u_6) (E : Type u_7) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Type u_7

The weak topology is the topology coarsest topology on E such that all functionals fun x => v x are continuous.

Equations

• WeakSpace 𝕜 E = WeakBilin (topDualPairing 𝕜 E).flip

instance WeakSpace.instAddCommMonoid {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : AddCommMonoid (WeakSpace 𝕜 E)

Equations

• WeakSpace.instAddCommMonoid = WeakBilin.instAddCommMonoid (topDualPairing 𝕜 E).flip

instance WeakSpace.instModule {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Module 𝕜 (WeakSpace 𝕜 E)

Equations

• WeakSpace.instModule = WeakBilin.instModule (topDualPairing 𝕜 E).flip

instance WeakSpace.instTopologicalSpace {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : TopologicalSpace (WeakSpace 𝕜 E)

Equations

• WeakSpace.instTopologicalSpace = WeakBilin.instTopologicalSpace (topDualPairing 𝕜 E).flip

instance WeakSpace.instContinuousAdd {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : ContinuousAdd (WeakSpace 𝕜 E)

instance WeakSpace.instModule' {𝕜 : Type u_2} {𝕝 : Type u_3} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [CommSemiring 𝕝] [Module 𝕝 E] : Module 𝕝 (WeakSpace 𝕜 E)

Equations

• WeakSpace.instModule' = WeakBilin.instModule' (topDualPairing 𝕜 E).flip

instance WeakSpace.instIsScalarTower {𝕜 : Type u_2} {𝕝 : Type u_3} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [CommSemiring 𝕝] [Module 𝕝 𝕜] [Module 𝕝 E] [IsScalarTower 𝕝 𝕜 E] : IsScalarTower 𝕝 𝕜 (WeakSpace 𝕜 E)

def WeakSpace.map {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F] (f : E →L[𝕜] F) : WeakSpace 𝕜 E →L[𝕜] WeakSpace 𝕜 F

A continuous linear map from E to F is still continuous when E and F are equipped with their weak topologies.

Equations

• WeakSpace.map f = { toLinearMap := ↑f, cont := ⋯ }

theorem WeakSpace.map_apply {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F] (f : E →L[𝕜] F) (x : E) : (map f) x = f x

@[simp]

theorem WeakSpace.coe_map {𝕜 : Type u_2} {E : Type u_4} {F : Type u_5} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F] (f : E →L[𝕜] F) : ⇑(map f) = ⇑f

def toWeakSpace (𝕜 : Type u_2) (E : Type u_4) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : E ≃ₗ[𝕜] WeakSpace 𝕜 E

There is a canonical map E → WeakSpace 𝕜 E (the "identity" mapping). It is a linear equivalence.

Equations

• toWeakSpace 𝕜 E = LinearEquiv.refl 𝕜 E

def toWeakSpaceCLM (𝕜 : Type u_2) (E : Type u_4) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : E →L[𝕜] WeakSpace 𝕜 E

For a topological vector space E, "identity mapping" E → WeakSpace 𝕜 E is continuous. This definition implements it as a continuous linear map.

Equations

• toWeakSpaceCLM 𝕜 E = { toLinearMap := ↑(toWeakSpace 𝕜 E), cont := ⋯ }

@[simp]

theorem toWeakSpaceCLM_eq_toWeakSpace (𝕜 : Type u_2) (E : Type u_4) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (x : E) : (toWeakSpaceCLM 𝕜 E) x = (toWeakSpace 𝕜 E) x

theorem toWeakSpaceCLM_bijective {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : Function.Bijective ⇑(toWeakSpaceCLM 𝕜 E)

theorem isOpenMap_toWeakSpace_symm {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] : IsOpenMap ⇑(toWeakSpace 𝕜 E).symm

The canonical map from WeakSpace 𝕜 E to E is an open map.

theorem WeakSpace.isOpen_of_isOpen {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (V : Set E) (hV : IsOpen (⇑(toWeakSpaceCLM 𝕜 E) '' V)) : IsOpen V

A set in E which is open in the weak topology is open.

theorem tendsto_iff_forall_eval_tendsto_topDualPairing {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] {l : Filter α} {f : α → WeakDual 𝕜 E} {x : WeakDual 𝕜 E} : Filter.Tendsto f l (nhds x) ↔ ∀ (y : E), Filter.Tendsto (fun (i : α) => ((topDualPairing 𝕜 E) (f i)) y) l (nhds (((topDualPairing 𝕜 E) x) y))

instance WeakSpace.instAddCommGroup {𝕜 : Type u_2} {E : Type u_4} [CommRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalAddGroup 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] : AddCommGroup (WeakSpace 𝕜 E)

Equations

• WeakSpace.instAddCommGroup = WeakBilin.instAddCommGroup (topDualPairing 𝕜 E).flip

instance WeakSpace.instIsTopologicalAddGroup {𝕜 : Type u_2} {E : Type u_4} [CommRing 𝕜] [TopologicalSpace 𝕜] [IsTopologicalAddGroup 𝕜] [ContinuousConstSMul 𝕜 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] : IsTopologicalAddGroup (WeakSpace 𝕜 E)