# mathlib3documentation

data.finsupp.pointwise

# The pointwise product on finsupp. #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

For the convolution product on finsupp when the domain has a binary operation, see the type synonyms add_monoid_algebra (which is in turn used to define polynomial and mv_polynomial) and monoid_algebra.

### Declarations about the pointwise product on finsupps #

@[protected, instance]
noncomputable def finsupp.has_mul {α : Type u₁} {β : Type u₂}  :
has_mul →₀ β)

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

Equations
theorem finsupp.coe_mul {α : Type u₁} {β : Type u₂} (g₁ g₂ : α →₀ β) :
(g₁ * g₂) = g₁ * g₂
@[simp]
theorem finsupp.mul_apply {α : Type u₁} {β : Type u₂} {g₁ g₂ : α →₀ β} {a : α} :
(g₁ * g₂) a = g₁ a * g₂ a
theorem finsupp.support_mul {α : Type u₁} {β : Type u₂} [decidable_eq α] {g₁ g₂ : α →₀ β} :
(g₁ * g₂).support g₁.support g₂.support
@[protected, instance]
noncomputable def finsupp.mul_zero_class {α : Type u₁} {β : Type u₂}  :
Equations
@[protected, instance]
noncomputable def finsupp.semigroup_with_zero {α : Type u₁} {β : Type u₂}  :
Equations
• finsupp.semigroup_with_zero = finsupp.semigroup_with_zero._proof_1 finsupp.semigroup_with_zero._proof_2 finsupp.semigroup_with_zero._proof_3
@[protected, instance]
noncomputable def finsupp.non_unital_non_assoc_semiring {α : Type u₁} {β : Type u₂}  :
Equations
• finsupp.non_unital_non_assoc_semiring = finsupp.non_unital_non_assoc_semiring._proof_1 finsupp.non_unital_non_assoc_semiring._proof_2 finsupp.non_unital_non_assoc_semiring._proof_3 finsupp.non_unital_non_assoc_semiring._proof_4 finsupp.non_unital_non_assoc_semiring._proof_5
@[protected, instance]
noncomputable def finsupp.non_unital_semiring {α : Type u₁} {β : Type u₂}  :
Equations
• finsupp.non_unital_semiring = finsupp.non_unital_semiring._proof_1 finsupp.non_unital_semiring._proof_2 finsupp.non_unital_semiring._proof_3 finsupp.non_unital_semiring._proof_4 finsupp.non_unital_semiring._proof_5
@[protected, instance]
noncomputable def finsupp.non_unital_comm_semiring {α : Type u₁} {β : Type u₂}  :
Equations
• finsupp.non_unital_comm_semiring = finsupp.non_unital_comm_semiring._proof_1 finsupp.non_unital_comm_semiring._proof_2 finsupp.non_unital_comm_semiring._proof_3 finsupp.non_unital_comm_semiring._proof_4 finsupp.non_unital_comm_semiring._proof_5
@[protected, instance]
noncomputable def finsupp.non_unital_non_assoc_ring {α : Type u₁} {β : Type u₂}  :
Equations
• finsupp.non_unital_non_assoc_ring = finsupp.non_unital_non_assoc_ring._proof_1 finsupp.non_unital_non_assoc_ring._proof_2 finsupp.non_unital_non_assoc_ring._proof_3 finsupp.non_unital_non_assoc_ring._proof_4 finsupp.non_unital_non_assoc_ring._proof_5 finsupp.non_unital_non_assoc_ring._proof_6 finsupp.non_unital_non_assoc_ring._proof_7 finsupp.non_unital_non_assoc_ring._proof_8
@[protected, instance]
noncomputable def finsupp.non_unital_ring {α : Type u₁} {β : Type u₂}  :
Equations
• finsupp.non_unital_ring = finsupp.non_unital_ring._proof_1 finsupp.non_unital_ring._proof_2 finsupp.non_unital_ring._proof_3 finsupp.non_unital_ring._proof_4 finsupp.non_unital_ring._proof_5 finsupp.non_unital_ring._proof_6 finsupp.non_unital_ring._proof_7 finsupp.non_unital_ring._proof_8
@[protected, instance]
noncomputable def finsupp.non_unital_comm_ring {α : Type u₁} {β : Type u₂}  :
Equations
• finsupp.non_unital_comm_ring = finsupp.non_unital_comm_ring._proof_1 finsupp.non_unital_comm_ring._proof_2 finsupp.non_unital_comm_ring._proof_3 finsupp.non_unital_comm_ring._proof_4 finsupp.non_unital_comm_ring._proof_5 finsupp.non_unital_comm_ring._proof_6 finsupp.non_unital_comm_ring._proof_7 finsupp.non_unital_comm_ring._proof_8
@[protected, instance]
noncomputable def finsupp.pointwise_scalar {α : Type u₁} {β : Type u₂} [semiring β] :
has_smul β) →₀ β)
Equations
@[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
@[protected, instance]
noncomputable def finsupp.pointwise_module {α : Type u₁} {β : Type u₂} [semiring β] :
module β) →₀ β)

The pointwise multiplicative action of functions on finitely supported functions

Equations