Theorems · Theorem · global analysis

HasFPowerSeriesWithinOnBall.fderivWithin_of_mem_of_analyticOn

∀ {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]
  {p : FormalMultilinearSeries 𝕜 E F} {r : ENNReal} {f : E → F} {x : E} {s : Set E},
  HasFPowerSeriesWithinOnBall f p s x r →
    AnalyticOn 𝕜 f s → UniqueDiffOn 𝕜 s → x ∈ s → HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f s) p.derivSeries s x r

If a function has a power series within a set on a ball, then so does its derivative. Version assuming that the function is analytic on s. For a version without this assumption but requiring that F is complete, see HasFPowerSeriesWithinOnBall.fderivWithin_of_mem.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

2 results in Mathlib

Foundations

Depth 190 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace