Theorems · Theorem · several complex variables

FormalMultilinearSeries.le_comp_radius_of_summable

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G)
  (p : FormalMultilinearSeries 𝕜 E F) (r : NNReal),
  (Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst) → ↑r ≤ (q.comp p).radius

Bounding below the radius of the composition of two formal multilinear series assuming summability over all compositions.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

1 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace