Theorems · Theorem · several complex variables

FormalMultilinearSeries.changeOriginSeries_finite_of_finite

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ},
  (∀ (m : ℕ), n ≤ m → p m = 0) → ∀ (k : ℕ) {m : ℕ}, n ≤ k + m → p.changeOriginSeries k m = 0

If p is a finite formal multilinear series, then so is p.changeOriginSeries k for every k in ℕ. More precisely, if p m = 0 for n ≤ m, then p.changeOriginSeries k m = 0 for n ≤ k + m.

Defined in

Mathlib.Analysis.Analytic.CPolynomialDef

Cited by

5 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace