# Category:Definitions/Supremum Metric

This category contains definitions related to Supremum Metric.
Related results can be found in Category:Supremum Metric.

Let $S$ be a set.

Let $M = \struct {A', d'}$ be a metric space.

Let $A$ be the set of all bounded mappings $f: S \to M$.

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

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

where $\sup$ denotes the supremum.

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