Theorems · Theorem · real analysis

ContinuousLinearMap.IsInvertible.contDiffAt_map_inverse

∀ {𝕜 : Type u_1} [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] {n : WithTop ℕ∞}
  [CompleteSpace E] {e : E →L[𝕜] F}, e.IsInvertible → ContDiffAt 𝕜 n ContinuousLinearMap.inverse e

At an invertible map e : M →L[R] M₂ between Banach spaces, the operation of inversion is C^n, for all n.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Operations

Cited by

2 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace