Theorems · Theorem · global analysis

ContinuousMultilinearMap.hasFTaylorSeriesUpTo_iteratedFDeriv

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [inst_3 : (i : ι) → NormedAddCommGroup (E i)]
  [inst_4 : (i : ι) → NormedSpace 𝕜 (E i)] [inst_5 : Fintype ι] (f : ContinuousMultilinearMap 𝕜 E F),
  HasFTaylorSeriesUpTo ⊤ ⇑f fun v n => f.iteratedFDeriv n v

A continuous multilinear function f admits a Taylor series, whose successive terms are given by f.iteratedFDeriv n. This is the point of the definition of f.iteratedFDeriv.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

1 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintype