Theorems · Theorem · global analysis

AnalyticOnNhd.fderiv

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type v} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {s : Set E} [CompleteSpace F], AnalyticOnNhd 𝕜 f s → AnalyticOnNhd 𝕜 (fderiv 𝕜 f) s

If a function is analytic on a set s, so is its Fréchet derivative. See also AnalyticOnNhd.fderiv_of_isOpen, removing the completeness assumption but requiring the set to be open.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

2 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace