Spaces with separating dual

We introduce a typeclass SeparatingDual R V, registering that the points of the topological module V over R can be separated by continuous linear forms.

This property is satisfied for normed spaces over ℝ or ℂ (by the analytic Hahn-Banach theorem) and for locally convex topological spaces over ℝ (by the geometric Hahn-Banach theorem).

We show in SeparatingDual.exists_ne_zero that given any non-zero vector in an R-module V satisfying SeparatingDual R V, there exists a continuous linear functional whose value on v is non-zero.

As a consequence of the existence of SeparatingDual.exists_ne_zero, a generalization of Hahn-Banach beyond the normed setting, we show that if V and W are nontrivial topological vector spaces over a topological field R that acts continuously on W, and if SeparatingDual R V, there are nontrivial continuous R-linear operators between V and W. This is recorded in the instance SeparatingDual.instNontrivialContinuousLinearMapIdOfContinuousSMul.

Under the assumption SeparatingDual R V, we show in SeparatingDual.exists_continuousLinearEquiv_apply_eq that the group of continuous linear equivalences acts transitively on the set of nonzero vectors.

class SeparatingDual (R : Type u_1) (V : Type u_2) [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] : Prop

When E is a topological module over a topological ring R, the class SeparatingDual R E registers that continuous linear forms on E separate points of E.

• exists_ne_zero' (x : V) : x ≠ 0 → ∃ (f : StrongDual R V), f x ≠ 0

  Any nonzero vector can be mapped by a continuous linear map to a nonzero scalar.

theorem separatingDual_def (R : Type u_1) (V : Type u_2) [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] : SeparatingDual R V ↔ ∀ (x : V), x ≠ 0 → ∃ (f : StrongDual R V), f x ≠ 0

instance instSeparatingDualRealOfIsTopologicalAddGroupOfContinuousSMulOfLocallyConvexSpaceOfT1Space {E : Type u_1} [TopologicalSpace E] [AddCommGroup E] [IsTopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] [T1Space E] : SeparatingDual ℝ E

instance instSeparatingDual {E : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] : SeparatingDual 𝕜 E

theorem SeparatingDual.exists_ne_zero {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] {x : V} (hx : x ≠ 0) : ∃ (f : StrongDual R V), f x ≠ 0

theorem SeparatingDual.exists_separating_of_ne {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] {x y : V} (h : x ≠ y) : ∃ (f : StrongDual R V), f x ≠ f y

theorem SeparatingDual.t1Space {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] [T1Space R] : T1Space V

theorem SeparatingDual.t2Space {R : Type u_1} {V : Type u_2} [Ring R] [AddCommGroup V] [TopologicalSpace V] [TopologicalSpace R] [Module R V] [SeparatingDual R V] [T2Space R] : T2Space V

theorem separatingDual_iff_injective {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [IsTopologicalRing R] [Module R V] : SeparatingDual R V ↔ Function.Injective ⇑(ContinuousLinearMap.coeLM R).flip

theorem SeparatingDual.dualMap_surjective_iff {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [IsTopologicalRing R] [Module R V] [SeparatingDual R V] {W : Type u_3} [AddCommGroup W] [Module R W] [FiniteDimensional R W] {f : W →ₗ[R] V} : Function.Surjective (⇑f.dualMap ∘ ContinuousLinearMap.toLinearMap) ↔ Function.Injective ⇑f

Given a finite-dimensional subspace W of a space V with separating dual, any linear functional on W extends to a continuous linear functional on V. This is stated more generally for an injective linear map from W to V.

instance SeparatingDual.instNontrivialContinuousLinearMapIdOfContinuousSMul {R : Type u_1} (V : Type u_2) [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [Module R V] [SeparatingDual R V] (W : Type u_3) [AddCommGroup W] [TopologicalSpace W] [Module R W] [Nontrivial W] [ContinuousSMul R W] [Nontrivial V] : Nontrivial (V →L[R] W)

theorem SeparatingDual.exists_eq_one {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [IsTopologicalRing R] [Module R V] [SeparatingDual R V] {x : V} (hx : x ≠ 0) : ∃ (f : StrongDual R V), f x = 1

theorem SeparatingDual.exists_eq_one_ne_zero_of_ne_zero_pair {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [IsTopologicalRing R] [Module R V] [SeparatingDual R V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : ∃ (f : StrongDual R V), f x = 1 ∧ f y ≠ 0

instance Algebra.IsCentral.continuousLinearMap {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [IsTopologicalRing R] [Module R V] [SeparatingDual R V] [IsTopologicalAddGroup V] [ContinuousSMul R V] : IsCentral R (V →L[R] V)

The center of continuous linear maps on a topological vector space with separating dual is trivial, in other words, it is a central algebra.

theorem SeparatingDual.exists_continuousLinearEquiv_apply_eq {R : Type u_1} {V : Type u_2} [Field R] [AddCommGroup V] [TopologicalSpace R] [TopologicalSpace V] [IsTopologicalRing R] [Module R V] [SeparatingDual R V] [IsTopologicalAddGroup V] [ContinuousSMul R V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) : ∃ (A : V ≃L[R] V), A x = y

In a topological vector space with separating dual, the group of continuous linear equivalences acts transitively on the set of nonzero vectors: given two nonzero vectors x and y, there exists A : V ≃L[R] V mapping x to y.

theorem SeparatingDual.completeSpace_of_completeSpace_continuousLinearMap (𝕜 : Type u_3) (E : Type u_4) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [SeparatingDual 𝕜 E] [Nontrivial E] [CompleteSpace (E →L[𝕜] F)] : CompleteSpace F

If a space of linear maps from E to F is complete, and E is nontrivial, then F is complete.

theorem SeparatingDual.completeSpace_continuousLinearMap_iff (𝕜 : Type u_3) (E : Type u_4) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [SeparatingDual 𝕜 E] [Nontrivial E] : CompleteSpace (E →L[𝕜] F) ↔ CompleteSpace F

theorem SeparatingDual.completeSpace_of_completeSpace_continuousMultilinearMap (𝕜 : Type u_3) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_6} [Finite ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] [∀ (i : ι), SeparatingDual 𝕜 (M i)] [CompleteSpace (ContinuousMultilinearMap 𝕜 M F)] {m : (i : ι) → M i} (hm : ∀ (i : ι), m i ≠ 0) : CompleteSpace F

If a space of multilinear maps from Π i, E i to F is complete, and each E i has a nonzero element, then F is complete.

theorem SeparatingDual.completeSpace_continuousMultilinearMap_iff (𝕜 : Type u_3) (F : Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_6} [Finite ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] [∀ (i : ι), SeparatingDual 𝕜 (M i)] {m : (i : ι) → M i} (hm : ∀ (i : ι), m i ≠ 0) : CompleteSpace (ContinuousMultilinearMap 𝕜 M F) ↔ CompleteSpace F