Theorems · Theorem · global analysis

ApproximatesLinearOn.to_inv

∀ {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}
  {f' : E ≃L[𝕜] F} {s : Set E} {c : NNReal} (hf : ApproximatesLinearOn f (↑f') s c)
  (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹),
  ApproximatesLinearOn (↑(hf.toPartialEquiv hc).symm) (↑f'.symm) (f '' s) (‖↑f'.symm‖₊ * (‖↑f'.symm‖₊⁻¹ - c)⁻¹ * c)

The inverse function is approximated linearly on f '' s by f'.symm.

Defined in

Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Cited by

1 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace