Definition:Pointwise Multiplication of Integer-Valued Functions

Definition
Let $S$ be a non-empty set. Let $f, g: S \to \Z$ be integer-valued functions.

Then the pointwise product of $f$ and $g$ is defined as:
 * $f \times g: S \to \Z:$
 * $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

where the $\times$ on the is integer multiplication.

Also denoted as
Using the other common notational forms for multiplication, this definition can also be written:
 * $\forall s \in S: \map {\paren {f \cdot g} } s := \map f s \cdot \map g s$

or:
 * $\forall s \in S: \map {\paren {f g} } s := \map f s \map g s$

Also see

 * Pointwise Multiplication on Integer-Valued Functions is Associative
 * Pointwise Multiplication on Integer-Valued Functions is Commutative


 * Pointwise multiplication is seen to be an instance of a pointwise operation on integer-valued functions.