Theorems · Theorem · functional analysis

LinearMap.IsWeak.continuous_of_continuous_eval

∀ {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [inst : CommSemiring 𝕜] [inst_1 : TopologicalSpace 𝕜]
  [inst_2 : AddCommMonoid E] [inst_3 : Module 𝕜 E] [inst_4 : AddCommMonoid F] [inst_5 : Module 𝕜 F]
  [inst_6 : TopologicalSpace E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) [hB : B.IsWeak] {α : Type u_7} [inst_7 : TopologicalSpace α]
  {f : α → E}, (∀ (y : F), Continuous fun x => (B (f x)) y) → Continuous f

A map f : α → E is continuous if all the maps fun a ↦ B (f a) y are continuous for each y : F.

Defined in

Mathlib.Topology.Algebra.Module.IsWeak

Cited by

1 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringTopologicalSpaceAddCommMonoidModuleAddCommMonoidModuleTopologicalSpaceLinearMap.IsWeakTopologicalSpace