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

instance Finsupp.instMulFinsuppToZero {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 : α →₀ β) (g₂ : α →₀ β) : ↑(g₁ * g₂) = ↑g₁ * ↑g₂

@[simp]

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

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

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

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

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

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

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

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

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

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

instance Finsupp.pointwiseScalar {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)

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