\begin{definition}[Definition:Pointwise Scalar Multiplication of Number-Valued Function]
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 $f: S \to \mathbb F$.
When one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$, the following definition arises:
The (binary) operation of '''pointwise scalar multiplication''' is defined on $\mathbb F \times \mathbb F^S$ as:
:$\times: \mathbb F \times \mathbb F^S \to \mathbb F^S: \forall \lambda \in \mathbb F, f \in \mathbb F^S:$
::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$
where the $\times$ on the {{RHS}} is conventional arithmetic multiplication.
This can be seen to be an instance of pointwise multiplication where one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$
\end{definition}