Theorems · Theorem · complex analysis

FormalMultilinearSeries.norm_mul_pow_le_of_lt_radius

∀ {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} [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) {r : NNReal}, ↑r < p.radius → ∃ C > 0, ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C

For r strictly smaller than the radius of p, then ‖pₙ‖ rⁿ is bounded.

Defined in

Mathlib.Analysis.Analytic.ConvergenceRadius

Cited by

6 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace