Definition:Supremum Metric

Definition
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 $A$.

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

The metric space $\struct {A, d}$ is denoted in some sources as:
 * $\map {\mathscr B} {X, M}$

but this notation is not universal.

Also see

 * Supremum Metric is Metric