Theorems · Definition · global analysis

HasStrictFDerivAt.toOpenPartialHomeomorph

{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} →
                    {a : E} → [CompleteSpace E] → HasStrictFDerivAt f (↑f') a → OpenPartialHomeomorph E F

Given a function with an invertible strict derivative at a, returns an OpenPartialHomeomorph with to_fun = f and a ∈ source. This is a part of the inverse function theorem. The other part HasStrictFDerivAt.to_localInverse states that the inverse function of this OpenPartialHomeomorph has derivative f'.symm.

Defined in

Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv

Cited by

16 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace