Definition:Pointwise Multiplication

Definition
Let $S$ be a non-empty set.

Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$.

Let $\mathbb F^S$ be the set of all mappings $f: S \to \mathbb F$.

The (binary) operation of pointwise multiplication is defined on $\mathbb F^S$ as:


 * $\times: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$
 * $\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

where the $\times$ on the is conventional arithmetic 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:
 * $\forall s \in S: \map {\paren {f g} } s := \map f s \, \map g s$

Specific Number Sets
Specific instantiations of this concept to particular number sets are as follows:

Also see

 * Pointwise Multiplication is Associative
 * Pointwise Multiplication is Commutative


 * Pointwise Addition

It can be seen that these definitions instantiate the more general Pointwise Operation on Number-Valued Functions.