English
There is a canonical additive monoid homomorphism from multilinear maps to the space of functions, given by evaluation: f ↦ (m ↦ f(m)). This map preserves addition and zero.
Русский
Существует каноническое аддитивное моноид-гомоморфизм от мультиленейных отображений к пространству функций, заданное отображением f ↦ (m ↦ f(m)); оно сохраняет сложение и ноль.
LaTeX
$$$\text{coeAddMonoidHom}: MultilinearMap R M_1 M_2 \to+ ( (\, (i: ι) → M_1 i) → M_2 )$ и является гомоморфизм-add$$
Lean4
/-- Equivalence between linear maps `M₂ →ₗ[R] M₃` and one-multilinear maps. -/
@[simps]
def ofSubsingleton [Subsingleton ι] (i : ι) : (M₂ →ₗ[R] M₃) ≃ MultilinearMap R (fun _ : ι ↦ M₂) M₃
where
toFun
f :=
{ toFun := fun x ↦ f (x i)
map_update_add' := by intros; simp [update_eq_const_of_subsingleton]
map_update_smul' := by intros; simp [update_eq_const_of_subsingleton] }
invFun
f :=
{ toFun := fun x ↦ f fun _ ↦ x
map_add' := fun x y ↦ by simpa [update_eq_const_of_subsingleton] using f.map_update_add 0 i x y
map_smul' := fun c x ↦ by simpa [update_eq_const_of_subsingleton] using f.map_update_smul 0 i c x }
right_inv f := by ext x; refine congr_arg f ?_; exact (eq_const_of_subsingleton _ _).symm
