Theorems · Theorem · global analysis

ApproximatesLinearOn.map_nhds_eq

∀ {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}
  [CompleteSpace E] {s : Set E} {c : NNReal} {f' : E →L[𝕜] F},
  ApproximatesLinearOn f f' s c →
    ∀ (f'symm : f'.NonlinearRightInverse) {x : E},
      s ∈ nhds x → Subsingleton F ∨ c < f'symm.nnnorm⁻¹ → Filter.map f (nhds x) = nhds (f x)

Defined in

Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Cited by

1 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace