# Definition:Supremum Metric/Bounded Real Functions on Interval

## Definition

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval.

Let $A$ be the set of all bounded real functions $f: \left[{a \,.\,.\, b}\right] \to \R$.

Let $d: A \times A \to \R$ be the function defined as:

$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in \left[{a \,.\,.\, b}\right]} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$

where $\sup$ denotes the supremum.

$d$ is known as the supremum metric on $A$.

## Also known as

This metric is also known as the sup metric or the uniform metric.

The metric space $\left({A, d}\right)$ is denoted in some sources as:

$\mathscr B \left({\left[{a \,.\,.\, b}\right], \R}\right)$

but this notation is not universal.