Theorems · Theorem · complex analysis

HasFPowerSeriesOnBall.r_eq_top_of_exists

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {f : 𝕜 → E} {r : ENNReal} {x : 𝕜} {p : FormalMultilinearSeries 𝕜 𝕜 E},
  HasFPowerSeriesOnBall f p x r →
    (∀ (r' : NNReal), 0 < r' → ∃ p', HasFPowerSeriesOnBall f p' x ↑r') → HasFPowerSeriesOnBall f p x ⊤

If a function f : 𝕜 → E has power series representation p on a ball of some radius and for each positive radius it has some power series representation, then p converges to f on the whole 𝕜.

Defined in

Mathlib.Analysis.Analytic.Uniqueness

Cited by

1 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace