Definition:Supremum Metric/Bounded Continuous Mappings

Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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:
 * $\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in A_1} d_2 \left({f \left({x}\right), g \left({x}\right)}\right)$

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.

Also see

 * Supremum Metric on Bounded Continuous Mappings is Metric