\begin{definition}[Definition:Pointwise Operation on Number-Valued Functions]
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 $f: S \to \mathbb F$.
Let $\oplus$ be a binary operation on $\mathbb F$.
The (binary) operation '''pointwise $\oplus$''' is defined on $\mathbb F^S$ as:
:$\oplus: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$
::$\forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$
\end{definition}