\begin{definition}[Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions]
Let $S$ be a non-empty set, and let ${\overline \R}^S$ be the set of all mappings $f: S \to \overline \R$.
Here $\overline \R$ denotes the extended set of real numbers.
Then '''pointwise ($\overline \R$-)scalar multiplication''' on ${\overline \R}^S$ is the binary operation $\cdot: \overline \R \times {\overline \R}^S \to {\overline \R}^S$ defined by:
:$\forall \lambda \in \overline \R: \forall f \in {\overline \R}^S: \forall s \in S: \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$
where the $\cdot$ on the right is extended real multiplication.
'''Pointwise scalar multiplication''' thence is an instance of a pointwise operation on extended real-valued functions.
\end{definition}