Theorems · Theorem · functional analysis

ContinuousAt.clm_apply

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : SeminormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  {X : Type u_6} [inst_5 : TopologicalSpace X] {f : X → E →L[𝕜] F} {g : X → E} {x : X},
  ContinuousAt f x → ContinuousAt g x → ContinuousAt (fun x => (f x) (g x)) x

Defined in

Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Cited by

1 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceTopologicalSpace