# Definition:Pointwise Operation on Extended Real-Valued Functions

## Definition

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.

## Examples

- Definition:Pointwise Addition of Extended Real-Valued Functions
- Definition:Pointwise Multiplication of Extended Real-Valued Functions
- Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions
- Definition:Absolute Value of Extended Real-Valued Function
- Definition:Pointwise Limit of Extended Real-Valued Functions
- Definition:Pointwise Maximum of Extended Real-Valued Functions
- Definition:Pointwise Minimum of Extended Real-Valued Functions

## 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.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 8$