Theorems · Theorem · global analysis

ApproximatesLinearOn.closedBall_subset_target

∀ {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] {ε : ℝ}
  {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : NNReal} [inst_5 : CompleteSpace E]
  (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹) (hs : IsOpen s) {b : E},
  0 ≤ ε →
    Metric.closedBall b ε ⊆ s →
      Metric.closedBall (f b) ((↑‖↑f'.symm‖₊⁻¹ - ↑c) * ε) ⊆
        (ApproximatesLinearOn.toOpenPartialHomeomorph f s hf hc hs).target

Defined in

Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Cited by

0 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace