Theorems · Theorem · functional analysis

ContinuousMultilinearMap.map_sum_finset

∀ {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [inst : Semiring R]
  [inst_1 : (i : ι) → AddCommMonoid (M₁ i)] [inst_2 : AddCommMonoid M₂] [inst_3 : (i : ι) → Module R (M₁ i)]
  [inst_4 : Module R M₂] [inst_5 : (i : ι) → TopologicalSpace (M₁ i)] [inst_6 : TopologicalSpace M₂]
  (f : ContinuousMultilinearMap R M₁ M₂) {α : ι → Type u_1} [inst_7 : Fintype ι] (g : (i : ι) → α i → M₁ i)
  (A : (i : ι) → Finset (α i)) [inst_8 : DecidableEq ι],
  (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ Fintype.piFinset A, f fun i => g i (r i)

If f is continuous multilinear, then f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ) is the sum of f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions with r 1 ∈ A₁, ..., r n ∈ Aₙ. This follows from multilinearity by expanding successively with respect to each coordinate.

Defined in

Mathlib.Topology.Algebra.Module.Multilinear.Basic

Cited by

1 results in Mathlib

Foundations

Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModuleTopologicalSpaceTopologicalSpaceFintypeDecidableEq