# Definition:Supremum Metric/Bounded Real-Valued Functions

## Definition

Let $S$ be a set.

Let $A$ be the set of all bounded real-valued functions $f: S \to \R$.

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

$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in S} \size {\map f x - \map g x}$

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.