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