Theorems · Theorem · real analysis

LinearIsometry.norm_iteratedFDerivWithin_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] {s : Set E} {x : E} {n : WithTop ℕ∞} {f : E → F}
  (g : F →ₗᵢ[𝕜] G),
  ContDiffWithinAt 𝕜 n f s x →
    UniqueDiffOn 𝕜 s →
      x ∈ s → ∀ {i : ℕ}, ↑i ≤ n → ‖iteratedFDerivWithin 𝕜 i (⇑g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖

Composition with a linear isometry on the left preserves the norm of the iterated derivative within a set.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Basic

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace