The pointwise product on Finsupp.

For the convolution product on Finsupp when the domain has a binary operation, see the type synonyms AddMonoidAlgebra (which is in turn used to define Polynomial and MvPolynomial) and MonoidAlgebra.

Declarations about the pointwise product on Finsupps

@[implicit_reducible]

noncomputable instance Finsupp.instMul {α : Type u₁} {β : Type u₂} [MulZeroClass β] : Mul (α →₀ β)

The product of f g : α →₀ β is the finitely supported function whose value at a is f a * g a.

theorem Finsupp.coe_mul {α : Type u₁} {β : Type u₂} [MulZeroClass β] (g₁ g₂ : α →₀ β) : ⇑(g₁ * g₂) = ⇑g₁ * ⇑g₂

@[simp]

theorem Finsupp.mul_apply {α : Type u₁} {β : Type u₂} [MulZeroClass β] {g₁ g₂ : α →₀ β} {a : α} : (g₁ * g₂) a = g₁ a * g₂ a

@[simp]

theorem Finsupp.single_mul {α : Type u₁} {β : Type u₂} [MulZeroClass β] (a : α) (b₁ b₂ : β) : (fun₀ | a => b₁ * b₂) = (fun₀ | a => b₁) * fun₀ | a => b₂

theorem Finsupp.support_mul_subset_left {α : Type u₁} {β : Type u₂} [MulZeroClass β] {g₁ g₂ : α →₀ β} : (g₁ * g₂).support ⊆ g₁.support

theorem Finsupp.support_mul_subset_right {α : Type u₁} {β : Type u₂} [MulZeroClass β] {g₁ g₂ : α →₀ β} : (g₁ * g₂).support ⊆ g₂.support

theorem Finsupp.support_mul {α : Type u₁} {β : Type u₂} [MulZeroClass β] [DecidableEq α] {g₁ g₂ : α →₀ β} : (g₁ * g₂).support ⊆ g₁.support ∩ g₂.support

@[implicit_reducible]

noncomputable instance Finsupp.instMulZeroClass {α : Type u₁} {β : Type u₂} [MulZeroClass β] : MulZeroClass (α →₀ β)

@[implicit_reducible]

noncomputable instance Finsupp.instSemigroupWithZero {α : Type u₁} {β : Type u₂} [SemigroupWithZero β] : SemigroupWithZero (α →₀ β)

@[implicit_reducible]

noncomputable instance Finsupp.instNonUnitalNonAssocSemiring {α : Type u₁} {β : Type u₂} [NonUnitalNonAssocSemiring β] : NonUnitalNonAssocSemiring (α →₀ β)

@[implicit_reducible]

noncomputable instance Finsupp.instNonUnitalSemiring {α : Type u₁} {β : Type u₂} [NonUnitalSemiring β] : NonUnitalSemiring (α →₀ β)

@[implicit_reducible]

noncomputable instance Finsupp.instNonUnitalCommSemiring {α : Type u₁} {β : Type u₂} [NonUnitalCommSemiring β] : NonUnitalCommSemiring (α →₀ β)

@[implicit_reducible]

noncomputable instance Finsupp.instNonUnitalNonAssocRing {α : Type u₁} {β : Type u₂} [NonUnitalNonAssocRing β] : NonUnitalNonAssocRing (α →₀ β)

@[implicit_reducible]

noncomputable instance Finsupp.instNonUnitalRing {α : Type u₁} {β : Type u₂} [NonUnitalRing β] : NonUnitalRing (α →₀ β)

@[implicit_reducible]

noncomputable instance Finsupp.instNonUnitalCommRing {α : Type u₁} {β : Type u₂} [NonUnitalCommRing β] : NonUnitalCommRing (α →₀ β)

theorem Finsupp.pointwise_smul_support_finite {α : Type u₁} {β : Type u₂} {γ : Type u₃} [Zero γ] [SMulZeroClass β γ] (f : α → β) (g : α →₀ γ) : (Function.support fun (x : α) => f x • g x).Finite

@[reducible, inline]

noncomputable abbrev Finsupp.pointwiseScalar {α : Type u₁} {β : Type u₂} {γ : Type u₃} [Zero γ] [SMulZeroClass β γ] : SMul (α → β) (α →₀ γ)

Pointwise scalar multiplication given by (f • g) x = f x • g x.

@[implicit_reducible]

noncomputable instance Finsupp.pointwiseScalarSemiring {α : Type u₁} {β : Type u₂} [Semiring β] : SMul (α → β) (α →₀ β)

@[simp]

theorem Finsupp.coe_pointwise_smul {α : Type u₁} {β : Type u₂} [Semiring β] (f : α → β) (g : α →₀ β) : ⇑(f • g) = f • ⇑g

@[implicit_reducible]

noncomputable instance Finsupp.pointwiseModule {α : Type u₁} {β : Type u₂} [Semiring β] : Module (α → β) (α →₀ β)

The pointwise multiplicative action of functions on finitely supported functions

instance Finsupp.instIsScalarTowerForall {α : Type u₁} {β : Type u₂} [Semiring β] : IsScalarTower β (α → β) (α →₀ β)