Theorems · Theorem · several complex variables

HasFiniteFPowerSeriesAt.eventually_const_of_bound_one

∀ {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}, HasFiniteFPowerSeriesAt f pf x 1 → f =ᶠ[nhds x] fun x_1 => f x

If f has a formal power series at x bounded by 1, then f is constant equal to f x in a neighborhood of x.

Defined in

Mathlib.Analysis.Analytic.CPolynomialDef

Cited by

0 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace