Theorems · Theorem · several complex variables

OpenPartialHomeomorph.hasFPowerSeriesAt_symm

∀ {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 : OpenPartialHomeomorph E F) {a : E} {i : E ≃L[𝕜] F},
  a ∈ f.source →
    ∀ {p : FormalMultilinearSeries 𝕜 E F},
      HasFPowerSeriesAt (↑f) p a →
        p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i → HasFPowerSeriesAt (↑f.symm) (p.leftInv i a) (↑f a)

If an open partial homeomorphism f is defined at a and has a power series expansion there with invertible linear term, then f.symm has a power series expansion at f a, given by the inverse of the initial power series.

Defined in

Mathlib.Analysis.Analytic.Inverse

Cited by

2 results in Mathlib

Foundations

Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace