Theorems · Theorem · several complex variables

HasFPowerSeriesAt.tendsto_partialSum_prod_of_comp

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {G : Type u_4}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f : E → G} {q : FormalMultilinearSeries 𝕜 F G}
  {p : FormalMultilinearSeries 𝕜 E F} {x : E},
  HasFPowerSeriesAt f (q.comp p) x →
    0 < q.radius →
      0 < p.radius →
        ∀ᶠ (y : E) in nhds 0,
          Filter.Tendsto (fun a => q.partialSum a.1 (p.partialSum a.2 y - (p 0) fun x => 0)) Filter.atTop
            (nhds (f (x + y)))

Defined in

Mathlib.Analysis.Analytic.Inverse

Cited by

1 results in Mathlib

Foundations

Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace