Theorems · Theorem · real analysis

hasFTaylorSeriesUpToOn_top_iff_right

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E}
  {f : E → F} {N : WithTop ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F},
  ↑⊤ ≤ N →
    (HasFTaylorSeriesUpToOn N f p s ↔
      (∀ x ∈ s, (p x 0).curry0 = f x) ∧
        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
          HasFTaylorSeriesUpToOn N (fun x => (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x => (p x).shift) s)

p is a Taylor series of f up to ⊤ if and only if p.shift is a Taylor series up to ⊤ for p 1, which is a derivative of f.

Defined in

Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Cited by

1 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace