Definition:Supremum Metric/Bounded Real Functions on Interval

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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 $A$.

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:

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

while some give it as:

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

There appears to be no standard notation for it.

Also presented as

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.

Also see

  • Results about the supremum metric can be found here.