Definition:Supremum Metric/Continuous Real Functions

Definition

Let $\closedint a b \subseteq \R$ be a closed real interval.

Let $A$ be the set of all continuous functions $f: \closedint a b \to \R$.

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

$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in \closedint a b} \size {\map f x - \map g x}$

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.

The metric space $\struct {A, d}$ is denoted in some sources as:

$\mathscr C \closedint a b$

but this notation is not universal.

Examples

Closure of $\map {B_1} 0$ on $\closedint 0 1$

Let $\closedint 0 1 \subseteq \R$ be the closed unit interval.

Let $\mathscr C \closedint 0 1$ be the supremum space of continuous functions $f: \closedint 0 1 \to \R$.

Then:

$\map \cl {\map {B_1} \bszero} = \set {f \in \mathscr C \closedint 0 1: \map {d_\infty} {f, \bszero} \le 1}$

where:

$\map {B_1} \bszero$ denotes the open $1$-ball of $\bszero$

$\d_\infty$ denotes the Chebyshev distance

$\bszero$ denotes the constant function $f_0$.

Also see

• Supremum Metric on Continuous Real Functions is Subspace of Bounded

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.8$