Definition:Pointwise Operation/Real-Valued Functions/Multiary

Definition
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 $\R$.

For ease of notation, write $\left[{S \to \R}\right]$ for $\R^S$.

Let $I$ be some index set.

Let $\oplus^I: \R^I \to \R$ be an $I$-ary 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:


 * $\forall \left({f_i}\right)_{i \mathop \in I} \in \left[{S \to \R}\right]^I: \forall s \in S: \left({\oplus^I \left({f_i}\right)_{i \in I} }\right) \left({s}\right) := \oplus^I \left({f_i \left({s}\right) }\right)_{i \in I}$