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

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.

Also see

 * Supremum Metric on Bounded Real-Valued Functions is Metric