Definition:Supremum Metric

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Definition

Let $S$ be a set.

Let $M = \left({A', d'}\right)$ be a metric space.

Let $A$ be the set of all bounded mappings $f: S \to M$.

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

$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in S} d' \left({f \left({x}\right), g \left({x}\right)}\right)$

where $\sup$ denotes the supremum.


$d$ is known as the supremum metric on $A$.


Special Cases

Bounded Continuous Mappings

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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:

$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in A_1} d_2 \left({f \left({x}\right), g \left({x}\right)}\right)$

where $\sup$ denotes the supremum.


$d$ is known as the supremum metric on $A$.


Bounded Real-Valued Functions

Let $S$ be a set.

Let $A$ be the set of all bounded real-valued functions $f: S \to \R$.

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

$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in S} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$

where $\sup$ denotes the supremum.


$d$ is known as the supremum metric on $A$.


Bounded Real Sequences

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

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

$\displaystyle \forall \left\langle{x_i}\right\rangle, \left\langle{y_i}\right\rangle \in A: d \left({\left\langle{x_i}\right\rangle, \left\langle{y_i}\right\rangle}\right) := \sup_{n \mathop \in \N} \left\vert{x_n - y_n}\right\vert$

where $\sup$ denotes the supremum.


$d$ is known as the supremum metric on $A$.


Bounded Real Functions on Interval

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval.

Let $A$ be the set of all bounded real functions $f: \left[{a \,.\,.\, b}\right] \to \R$.

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

$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in \left[{a \,.\,.\, b}\right]} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$

where $\sup$ denotes the supremum.


$d$ is known as the supremum metric on $A$.


Continuous Real Functions

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval.

Let $A$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$.

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

$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in \left[{a \,.\,.\, b}\right]} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$

where $\sup$ denotes the supremum.


$d$ is known as the supremum metric on $A$.


Differentiability Class

Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval.

Let $r \in \N$ be a natural number.

Let $\mathscr D^r \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$ which are of differentiability class $r$.

Let $d: \mathscr D^r \left[{a \,.\,.\, b}\right] \times \mathscr D^r \left[{a \,.\,.\, b}\right] \to \R$ be the function defined as:

$\displaystyle \forall f, g \in \mathscr D^r \left[{a \,.\,.\, b}\right]: d \left({f, g}\right) := \sup_{\substack {x \mathop \in \left[{a \,.\,.\, b}\right] \\ i \in \left\{ {0, 1, 2, \ldots, r}\right\} } } \left\vert{f^{\left({i}\right)} \left({x}\right) - g^{\left({i}\right)} \left({x}\right)}\right\vert$

where:

$f^{\left({i}\right)}$ denotes the $i$th derivative of $f$
$f^{\left({0}\right)}$ denotes $f$
$\sup$ denotes the supremum.

$d$ is known as the supremum metric on $\mathscr D^r \left[{a \,.\,.\, b}\right]$.


Also known as

This metric is also known as the sup metric or the uniform metric.

The metric space $\left({A, d}\right)$ is denoted in some sources as:

$\mathscr B \left({X, M}\right)$

but this notation is not universal.


Also see

  • Results about the supremum metric can be found here.


Sources