Theorems · Theorem · functional analysis

WeakDual.continuous_of_continuous_eval_re

∀ {α : Type u_4} {𝕜 : Type u_5} {F : Type u_7} [inst : TopologicalSpace α] [inst_1 : RCLike 𝕜] [inst_2 : AddCommGroup F]
  [inst_3 : Module 𝕜 F] [inst_4 : TopologicalSpace F] {g : α → WeakDual 𝕜 F},
  (∀ (x : F), Continuous fun a => RCLike.re ((g a) x)) → Continuous g

A map into WeakDual 𝕜 F over 𝕜 (with RCLike 𝕜) is continuous if the real parts of all the evaluation maps a ↦ g a y are continuous for each y : F.

Defined in

Mathlib.Analysis.Normed.Module.WeakDual

Cited by

0 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceRCLikeAddCommGroupModuleTopologicalSpace