\begin{definition}[Definition:Pointwise Multiplication]
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 {{RHS}} is conventional arithmetic multiplication.
\end{definition}