Theorems · Theorem · several complex variables

HasFPowerSeriesWithinOnBall.tendstoUniformlyOn

∀ {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} {r' : NNReal},
  HasFPowerSeriesWithinOnBall f p s x r →
    ↑r' < r →
      TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) Filter.atTop
        ((fun x_1 => x + x_1) ⁻¹' insert x s ∩ Metric.ball 0 ↑r')

If a function admits a power series expansion within a set at x, then it is the uniform limit of the partial sums of this power series on strict subdisks of the disk of convergence, i.e., f (x + y) is the uniform limit of p.partialSum n y there.

Defined in

Mathlib.Analysis.Analytic.Basic

Cited by

3 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace