Theorems · Theorem · real analysis

contDiffOn_clm_apply

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {D : Type uD} [inst_1 : NormedAddCommGroup D]
  [inst_2 : NormedSpace 𝕜 D] {E : Type uE} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] {F : Type uF}
  [inst_5 : NormedAddCommGroup F] [inst_6 : NormedSpace 𝕜 F] {n : WithTop ℕ∞} [CompleteSpace 𝕜] {f : D → E →L[𝕜] F}
  {s : Set D} [FiniteDimensional 𝕜 E], ContDiffOn 𝕜 n f s ↔ ∀ (y : E), ContDiffOn 𝕜 n (fun x => (f x) y) s

A family of continuous linear maps is C^n on s if all its applications are.

Defined in

Mathlib.Analysis.Calculus.ContDiff.FiniteDimension

Cited by

2 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceFiniteDimensional