Theorems · Theorem · several complex variables

HasFiniteFPowerSeriesOnBall.cpolynomialAt_of_mem

∀ {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} {r : ENNReal} {n : ℕ} {x y : E},
  HasFiniteFPowerSeriesOnBall f p x n r → y ∈ Metric.eball x r → CPolynomialAt 𝕜 f y

If a function admits a finite power series expansion p on an open ball B (x, r), then it is continuously polynomial at every point of this ball.

Defined in

Mathlib.Analysis.Analytic.CPolynomialDef

Cited by

5 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace