Theorems · Theorem · functional analysis

continuousWithinAt_clm_apply

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type v} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type w} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace 𝕜]
  {X : Type u_1} [inst_6 : TopologicalSpace X] [FiniteDimensional 𝕜 E] {f : X → E →L[𝕜] F} {s : Set X} {x : X},
  ContinuousWithinAt f s x ↔ ∀ (y : E), ContinuousWithinAt (fun q => (f q) y) s x

A family of continuous linear maps is continuous within s at x iff all its applications are.

Defined in

Mathlib.Analysis.Normed.Module.FiniteDimension

Cited by

0 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceTopologicalSpaceFiniteDimensional