Theorems · Theorem · functional analysis

ContinuousMultilinearMap.map_smul_univ

∀ {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} [inst : CommSemiring 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) [inst_7 : Fintype ι] (c : ι → R) (m : (i : ι) → M₁ i),
  (f fun i => c i • m i) = (∏ i, c i) • f m

Multiplicativity of a continuous multilinear map along all coordinates at the same time, writing f (fun i ↦ c i • m i) as (∏ i, c i) • f m.

Defined in

Mathlib.Topology.Algebra.Module.Multilinear.Basic

Cited by

5 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModuleTopologicalSpaceTopologicalSpaceFintype