Theorems · Theorem · several complex variables

FormalMultilinearSeries.compAlongComposition_norm

∀ {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] {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G)
  (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n),
  ‖q.compAlongComposition p c‖ ≤ ‖q c.length‖ * ∏ i, ‖p (c.blocksFun i)‖

The norm of q.compAlongComposition p c is controlled by the product of the norms of the relevant bits of q and p.

Defined in

Mathlib.Analysis.Analytic.Composition

Cited by

2 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace