# Definition:Supremum Metric/Bounded Continuous Mappings

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## Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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:

- $\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in A_1} \map {d_2} {\map f x, \map g x}$

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**.

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.17$