Theorems · Theorem · several complex variables

FormalMultilinearSeries.comp_partialSum

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G)
  (p : FormalMultilinearSeries 𝕜 E F) (M N : ℕ) (z : E),
  q.partialSum M (∑ i ∈ Finset.Ico 1 N, (p i) fun _j => z) =
    ∑ i ∈ FormalMultilinearSeries.compPartialSumTarget 0 M N, (q.compAlongComposition p i.snd) fun _j => z

Composing the partial sums of two multilinear series coincides with the sum over all compositions in compPartialSumTarget 0 N N. This is precisely the motivation for the definition of compPartialSumTarget.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

2 results in Mathlib

Foundations

Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace