Theorems · Theorem · several complex variables

HasFiniteFPowerSeriesOnBall.eq_zero_of_bound_zero

∀ {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}
  {pf : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ENNReal},
  HasFiniteFPowerSeriesOnBall f pf x 0 r → ∀ y ∈ Metric.eball x r, f y = 0

If f has a formal power series on a ball bounded by 0, then f is equal to 0 on the ball.

Defined in

Mathlib.Analysis.Analytic.CPolynomialDef

Cited by

2 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace