Theorems · Theorem · several complex variables

HasFiniteFPowerSeriesOnBall.eq_partialSum

∀ {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} {x : E} {r : ENNReal} {n : ℕ},
  HasFiniteFPowerSeriesOnBall f p x n r → ∀ y ∈ Metric.eball 0 r, ∀ (m : ℕ), n ≤ m → f (x + y) = p.partialSum m y

If a function admits a finite power series expansion bounded by n, then it is equal to the mth partial sums of this power series at every point of the disk for n ≤ m.

Defined in

Mathlib.Analysis.Analytic.CPolynomialDef

Cited by

1 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace