English
There is a canonical left decomposition when exactly one component is nonzero: dfinsuppFamily (Pi.single p f) equals (DFinsupp.lsingle p) composed with a suitable left-linearization of f.
Русский
Существует каноническое левое разложение, когда ровно один компонент не ноль: dfinsuppFamily (Pi.single p f) эквивалентно композиции (DFinsupp.lsingle p) и левой линейной обработкой f.
LaTeX
$$$\dfinsuppFamily(\Pi.single p f) = (\DFinsupp.lsingle p)\circ_{\mathrm{lin}} (f \circ_{\mathrm{lin}} (\DFinsupp.lapply (p i))).$$$
Lean4
/-- `MultilinearMap.dfinsuppFamily` as a linear map. -/
@[simps]
def dfinsuppFamilyₗ :
(Π (p : Π i, κ i), MultilinearMap R (fun i ↦ M i (p i)) (N p)) →ₗ[R]
MultilinearMap R (fun i => Π₀ j : κ i, M i j) (Π₀ t : Π i, κ i, N t)
where
toFun := dfinsuppFamily
map_add' := dfinsuppFamily_add
map_smul' := dfinsuppFamily_smul
