Definition:Pointwise Operation on Extended Real-Valued Functions

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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:

$\map {\oplus^I \family {f_i}_{i \mathop \in I} } s := \oplus^I \family {\map {f_i} s}_{i \mathop \in I}$

Also known as

When $\oplus$ or $\oplus^I$ has a distinguished name, it is usual to name the corresponding $\oplus$ on functions by prepending pointwise to that name.


Also see

It can be seen that these definitions instantiate the general induced operation on set of mappings.

Compare also the definition for real-valued functions.