Theorems · Theorem · global analysis

AnalyticWithinAt.exists_hasFTaylorSeriesUpToOn

∀ {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} {x : E}
  {s : Set E} [CompleteSpace F] (n : WithTop ℕ∞),
  AnalyticWithinAt 𝕜 f s x →
    ∃ u ∈ nhdsWithin x (insert x s), ∃ p, HasFTaylorSeriesUpToOn n f p u ∧ ∀ (i : ℕ), AnalyticOn 𝕜 (fun x => p x i) u

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

1 results in Mathlib

Foundations

Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace