\begin{definition}[Definition:Pointwise Operation/Real-Valued Functions]
Let $S$ be a non-empty set.
Let $\R^S$ be the set of all mappings $f: S \to \R$, where $\R$ is the set of real numbers.
Let $\oplus$ be a binary operation on $\R$.
Define $\oplus: \R^S \times \R^S \to \R^S$, called '''pointwise $\oplus$''', by:
:$\forall f, g \in \R^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$
In the above expression, the operator on the {{RHS}} is the given $\oplus$ on the real numbers.
\end{definition}