Theorems · Theorem · real analysis

iteratedFDerivWithin_succ_eq_comp_left

∀ {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 : ℕ},
  iteratedFDerivWithin 𝕜 (n + 1) f s =
    ⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F).symm ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s

Writing explicitly the n+1-th derivative as the composition of a currying linear equiv, and the derivative of the n-th derivative.

Defined in

Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Cited by

7 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace