# Definition:Supremum Metric/Bounded Continuous Mappings

## Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $A$ be the set of all continuous mappings $f: M_1 \to M_2$ which are also bounded.

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

$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in A_1} \map {d_2} {\map f x, \map g x}$

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.