Theorems · Theorem · real analysis

analyticOn_taylorComp

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {G : Type u_4}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {s : Set E} {t : Set F}
  {q : F → FormalMultilinearSeries 𝕜 F G} {p : E → FormalMultilinearSeries 𝕜 E F},
  (∀ (n : ℕ), AnalyticOn 𝕜 (fun x => q x n) t) →
    (∀ (n : ℕ), AnalyticOn 𝕜 (fun x => p x n) s) →
      ∀ {f : E → F},
        AnalyticOn 𝕜 f s → Set.MapsTo f s t → ∀ (n : ℕ), AnalyticOn 𝕜 (fun x => (q (f x)).taylorComp (p x) n) s

Defined in

Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace