Theorems · Theorem · functional analysis

LinearEquiv.image_closure_of_convex

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
  [inst_3 : AddCommGroup F] [inst_4 : Module 𝕜 F] [inst_5 : Module ℝ E] [IsScalarTower ℝ 𝕜 E] [inst_7 : Module ℝ F]
  [IsScalarTower ℝ 𝕜 F] [inst_9 : TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E]
  [LocallyConvexSpace ℝ E] [inst_13 : TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜 F]
  [LocallyConvexSpace ℝ F] {s : Set E},
  Convex ℝ s →
    ∀ (e : E ≃ₗ[𝕜] F),
      (∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)) →
        (∀ (f : StrongDual 𝕜 E), Continuous ⇑(e.symm.dualMap ↑f)) → ⇑e '' closure s = closure (⇑e '' s)

If e is a linear isomorphism between two locally convex spaces, and e induces (via precomposition) an isomorphism between their continuous duals, then e commutes with the closure on convex sets. The hypotheses hold automatically for e := toWeakSpace 𝕜 E, see Convex.toWeakSpace_closure.

Defined in

Mathlib.Analysis.LocallyConvex.WeakSpace

Cited by

1 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeAddCommGroupModuleAddCommGroupModuleModuleIsScalarTowerModuleIsScalarTowerTopologicalSpaceIsTopologicalAddGroupContinuousSMulLocallyConvexSpaceTopologicalSpaceIsTopologicalAddGroupContinuousSMulLocallyConvexSpace