Theorems · Theorem · several complex variables

FormalMultilinearSeries.rightInv.eq_def

∀ {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) (x_1 : ℕ),
  p.rightInv i x x_1 =
    match x_1 with
    | 0 => ContinuousMultilinearMap.uncurry0 𝕜 F x
    | 1 => (continuousMultilinearCurryFin1 𝕜 F E).symm ↑i.symm
    | n.succ.succ =>
      have q := fun k => if k < n + 2 then p.rightInv i x k else 0;
      -(↑i.symm).compContinuousMultilinearMap (p.comp q (n + 2))

Defined in

Mathlib.Analysis.Analytic.Inverse

Cited by

0 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace