comp_continuousLinearMap — Mathlib

English

Composition with a linear isomorphism on the right preserves the norm of the iterated derivative.

Русский

Композиция справа линейной изоморфизмом сохраняет норму итеративной производной.

LaTeX

$$$\forall g:\, G \toL[\mathbb{K}] E,\; \|iteratedFDerivWithin 𝕜 i (f\circ g) (g^{-1} s) x\| = \|iteratedFDerivWithin 𝕜 i f s (g x)\|$$$

Lean4

/-- Composition by continuous linear maps on the right preserves `C^n` functions. -/
theorem comp_continuousLinearMap {f : E → F} {g : G →L[𝕜] E} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n (f ∘ g) :=
  contDiffOn_univ.1 <| ContDiffOn.comp_continuousLinearMap (contDiffOn_univ.2 hf) _

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