Theorems · Theorem · several complex variables

FormalMultilinearSeries.radius_leftInv_pos_of_radius_pos

∀ {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},
  0 < p.radius → p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i → 0 < (p.leftInv i x).radius

If a a formal multilinear series has a positive radius of convergence, then its left inverse also has a positive radius of convergence.

Defined in

Mathlib.Analysis.Analytic.Inverse

Cited by

1 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace