English
There is a natural linear equivalence between families of continuous multilinear maps indexed by i and a single multilinear map valued in the function type i ↦ M' i.
Русский
Существует естественное линейное биекое отображение между семействами непрерывных мультилінійних отображений и единым отображением в функциятом типе.
LaTeX
$$piLinearEquiv : (∀ i, ContinuousMultilinearMap A M₁ (M' i)) ≃ₗ[R'] ContinuousMultilinearMap A M₁ (∀ i, M' i)$$
Lean4
/-- `ContinuousMultilinearMap.pi` as a `LinearEquiv`. -/
@[simps +simpRhs]
def piLinearEquiv {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)] [∀ i, TopologicalSpace (M' i)]
[∀ i, ContinuousAdd (M' i)] [∀ i, Module R' (M' i)] [∀ i, Module A (M' i)] [∀ i, SMulCommClass A R' (M' i)]
[∀ i, ContinuousConstSMul R' (M' i)] :
(∀ i, ContinuousMultilinearMap A M₁ (M' i)) ≃ₗ[R'] ContinuousMultilinearMap A M₁ (∀ i, M' i) :=
{ piEquiv with
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl }
