Definition:Supremum Metric/Bounded Real Functions on Interval

Definition

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

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

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:

$\map {\mathscr B} {\closedint a b, \R}$

while some give it as:

$\map {\mathscr C} {\closedint a b}$

There appears to be no standard notation for it.

When Bert Mendelson introduces this metric in his Introduction to Topology, 3rd ed. of $1975$, he uses the notation:

$\map {d'} {f, g} = \operatorname {l.u.b.} \bigcup_{x \in \sqbrk {a, b} } \set {\size {\map f x, \map g x} }$

which seems to serve no purpose except to unnecessarily overcomplicate the notation.

It is effectively using $\bigcup_{a \mathop \in A} \set a$ for $A$, expressing it as the union of singletons.

1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Exercise $5$
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.8$