Theorems · Theorem · complex analysis

hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {F : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F}
  {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E},
  HasFPowerSeriesWithinAt f p s x ↔ ∃ g, f =ᶠ[nhdsWithin x (insert x s)] g ∧ HasFPowerSeriesAt g p x

f has power series p at x iff some local extension of f has that series

Defined in

Mathlib.Analysis.Analytic.Within

Cited by

0 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace