Theorems · Theorem · several complex variables

FormalMultilinearSeries.comp_rightInv

∀ {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]
  (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E),
  p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i →
    p.comp (p.rightInv i x) = FormalMultilinearSeries.id 𝕜 F ((p 0) 0)

The right inverse to a formal multilinear series is indeed a right inverse, provided its linear term is invertible and its constant term vanishes.

Defined in

Mathlib.Analysis.Analytic.Inverse

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace