Theorems · Theorem · several complex variables

HasFPowerSeriesWithinOnBall.hasSum

∀ {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] {f : E → F}
  {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ENNReal},
  HasFPowerSeriesWithinOnBall f p s x r →
    ∀ {y : E}, x + y ∈ insert x s → y ∈ Metric.eball 0 r → HasSum (fun n => (p n) fun x => y) (f (x + y))

p converges to f within s

Defined in

Mathlib.Analysis.Analytic.Basic

Cited by

28 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace