\begin{definition}[Definition:Pointwise Scalar Multiplication of Mappings]
Let $\struct {R, +_R, \times_R}$ be a ring, and let $\struct {S, \circ}_R$ be an $R$-algebraic structure.
Let $X$ be a non-empty set, and let $S^X$ be the set of all mappings from $X$ to $S$.
Then '''pointwise ($R$)-scalar multiplication''' on $S^X$ is the binary operation $\circ: R \times S^X \to S^X$ (the $\circ$ is the same as for $S$) defined by:
:$\forall \lambda \in R: \forall f \in S^X: \forall x \in X: \map {\paren {\lambda \circ f} } x := \lambda \circ \map f x$
The double use of $\circ$ is justified as $\struct {S^X, \circ}_R$ inherits all abstract-algebraic properties $\struct {S, \circ}_R$ might have.
This is rigorously formulated and proved on Mappings to R-Algebraic Structure form Similar R-Algebraic Structure.
\end{definition}