Theorems · Theorem · real analysis

ContinuousLinearMap.iteratedFDeriv_comp_left

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {x : E} {n : WithTop ℕ∞} {f : E → F} (g : F →L[𝕜] G),
  ContDiffAt 𝕜 n f x →
    ∀ {i : ℕ}, ↑i ≤ n → iteratedFDeriv 𝕜 i (⇑g ∘ f) x = g.compContinuousMultilinearMap (iteratedFDeriv 𝕜 i f x)

The iterated derivative of the composition with a linear map on the left is obtained by applying the linear map to the iterated derivative.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Basic

Cited by

4 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace