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 $\sqbrk {S \to \R}$ for $\R^S$.

Let $I$ be some indexing set.

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

Then $\oplus^I: \sqbrk {S \to \R}^I \to \sqbrk {S \to \R}$, referred to as pointwise $\oplus^I$, is defined as:

$\forall \family {f_i}_{i \mathop \in I} \in \sqbrk {S \to \R}^I: \forall s \in S: \map {\paren {\oplus^I \family {f_i}_{i \mathop \in I} } } s := \oplus^I \family {\map {f_i} s}_{i \mathop \in I}$