Definition:Supremum Metric

Definition
Let $S$ be a set.

Let $M = \left({A', d'}\right)$ 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:
 * $\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in S} d' \left({f \left({x}\right), g \left({x}\right)}\right)$

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:
 * $\mathscr B \left({X, M}\right)$

but this notation is not universal.

Also see

 * Supremum Metric is Metric