Theorems · Theorem · real analysis

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), ContDiffAt 𝕜 n ContinuousLinearMap.inverse ↑e

At a continuous linear equivalence e : E ≃L[𝕜] F between Banach spaces, the operation of inversion is C^n, for all n.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Operations

Cited by

3 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace