Theorems · Theorem · functional analysis

ContinuousMultilinearMap.eq_prod_iff

∀ {R : Type u} {ι : Type v} {M₁ : ι → Type w₁} {M₂ : Type w₂} {M₃ : Type w₃} [inst : Semiring R]
  [inst_1 : (i : ι) → AddCommMonoid (M₁ i)] [inst_2 : AddCommMonoid M₂] [inst_3 : AddCommMonoid M₃]
  [inst_4 : (i : ι) → Module R (M₁ i)] [inst_5 : Module R M₂] [inst_6 : Module R M₃]
  [inst_7 : (i : ι) → TopologicalSpace (M₁ i)] [inst_8 : TopologicalSpace M₂] [inst_9 : TopologicalSpace M₃]
  {f : ContinuousMultilinearMap R M₁ (M₂ × M₃)} {g : ContinuousMultilinearMap R M₁ M₂}
  {h : ContinuousMultilinearMap R M₁ M₃},
  f = g.prod h ↔
    (ContinuousLinearMap.fst R M₂ M₃).compContinuousMultilinearMap f = g ∧
      (ContinuousLinearMap.snd R M₂ M₃).compContinuousMultilinearMap f = h

Defined in

Mathlib.Topology.Algebra.Module.Multilinear.Basic

Cited by

0 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleTopologicalSpaceTopologicalSpaceTopologicalSpace