Theorems · Theorem · global analysis

ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse

∀ {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) {ε : ℝ} {b : E},
      0 ≤ ε →
        Metric.closedBall b ε ⊆ s →
          Set.SurjOn f (Metric.closedBall b ε) (Metric.closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε))

If a function is linearly approximated by a continuous linear map with a (possibly nonlinear) right inverse, then it is locally onto: a ball of an explicit radius is included in the image of the map.

Defined in

Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Cited by

3 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace