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