Theorems · Theorem · complex analysis

AnalyticOn.iteratedFDerivWithin_comp_perm

∀ {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] {f : E → F}
  {s : Set E} {x : E},
  AnalyticOn 𝕜 f s →
    UniqueDiffOn 𝕜 s →
      x ∈ s →
        ∀ {n : ℕ} (v : Fin n → E) (σ : Equiv.Perm (Fin n)),
          (iteratedFDerivWithin 𝕜 n f s x) (v ∘ ⇑σ) = (iteratedFDerivWithin 𝕜 n f s x) v

The n-th iterated derivative of an analytic function on a set is symmetric.

Defined in

Mathlib.Analysis.Analytic.IteratedFDeriv

Cited by

3 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace