\begin{definition}[Definition:Pointwise Operation on Extended Real-Valued Functions]
Let $S$ be a set, and let $f, g : S \to \overline \R$ be extended real-valued functions.
Let $\lambda \in \R$.
Then extended real-valued functions can be formed by defining (for all $s \in S$):
:$\lambda \cdot f: S \to \R, \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$
:$f + g: S \to \R, \map {\paren {f + g} } s := \map f s + \map g s$
:$f \cdot g: S \to \R, \map {\paren {f \cdot g} } s := \map f s \cdot \map g s$
as is done on Pointwise Scalar Multiplication, Pointwise Addition and Pointwise Multiplication, respectively.
More generally, let $\oplus$ be a binary operation on $\overline \R$.
Define $f \oplus g: S \to \overline \R$, called '''pointwise $\oplus$''', by:
:$\map {\paren {f \oplus g} } s := \map f s \oplus \map g s$
In above expressions, the subscript ${\overline \R}$ of an operator expresses that its operands are extended real numbers.
Next, let $\family {f_i}_{i \mathop \in I}, f_i: S \to \overline \R$, be any $I$-indexed collection of extended real-valued functions, where $I$ is some index set.
Suppose that $\oplus^I$ is an $I$-ary operation on $\overline \R$.
Then define $\oplus^I \family {f_i}_{i \mathop \in I}: S \to \overline \R$, called '''pointwise $\oplus^I$''', by:
:$\map {\oplus^I \family {f_i}_{i \mathop \in I} } s := \oplus^I \family {\map {f_i} s}_{i \mathop \in I}$
\end{definition}