Definition:Supremum Metric/Bounded Real Sequences

Definition

Let $A$ be the set of all bounded real sequences.

Let $d: A \times A \to \R$ be the function defined as:

$\ds \forall \sequence {x_i}, \sequence {y_i} \in A: \map d {\sequence {x_i}, \sequence {y_i} } := \sup_{n \mathop \in \N} \size {x_n - y_n}$

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 Sequences is Metric

Sources

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