linearDeriv — Mathlib

English

Applying domDomRestrict to arguments yields the expected coordinatewise relation with if-then definitions.

Русский

Применение domDomRestrict к аргументам даёт ожидаемую по координатам связь через условия if-then.

LaTeX

$$theorem domDomRestrict_apply (f) (P) (x) (z) : f.domDomRestrict P z x = f (fun j => if P j then x j else z j)$$

Lean4

/-- The "derivative" of a multilinear map, as a linear map from `(i : ι) → M₁ i` to `M₂`.
For continuous multilinear maps, this will indeed be the derivative. -/
def linearDeriv [DecidableEq ι] [Fintype ι] (f : MultilinearMap R M₁ M₂) (x : (i : ι) → M₁ i) :
    ((i : ι) → M₁ i) →ₗ[R] M₂ :=
  ∑ i : ι, (f.toLinearMap x i).comp (LinearMap.proj i)

Похожие утверждения