Theorems · Theorem · several complex variables

HasFPowerSeriesWithinOnBall.pi

∀ {𝕜 : Type u_2} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {ι : Type u_9} [inst_3 : Fintype ι] {e : E} {Fm : ι → Type u_10}
  [inst_4 : (i : ι) → NormedAddCommGroup (Fm i)] [inst_5 : (i : ι) → NormedSpace 𝕜 (Fm i)] {f : (i : ι) → E → Fm i}
  {s : Set E} {r : ENNReal} {p : (i : ι) → FormalMultilinearSeries 𝕜 E (Fm i)},
  (∀ (i : ι), HasFPowerSeriesWithinOnBall (f i) (p i) s e r) →
    0 < r → HasFPowerSeriesWithinOnBall (fun x x_1 => f x_1 x) (FormalMultilinearSeries.pi p) s e r

If each function in a finite family has a power series within a ball, then so does the family as a whole. Note that the positivity assumption on the radius is only needed when the family is empty.

Defined in

Mathlib.Analysis.Analytic.Constructions

Cited by

3 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceFintypeNormedAddCommGroupNormedSpace