\begin{definition}[Definition:Pointwise Operation/Real-Valued Functions/Multiary]
For ease of notation, write $\left[{S \to \R}\right]$ for $\R^S$.
Let $I$ be some [[Definition:Index Set|index set]].
Let $\oplus^I: \R^I \to \R$ be an [[Definition:Arity|$I$-ary]] [[Definition:Operation|operation]] on $\R$.
Then $\oplus^I: \left[{S \to \R}\right]^I \to \left[{S \to \R}\right]$, referred to as '''pointwise $\oplus^I$''', is defined as:
:$\forall \left({f_i}\right)_{i \mathop \in I} \in \left[{S \to \R}\right]^I: \forall s \in S: \left({\oplus^I \left({f_i}\right)_{i \in I} }\right) \left({s}\right) := \oplus^I \left({f_i \left({s}\right) }\right)_{i \in I}$
\end{definition}
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