Definition:Supremum Metric/Differentiability Class

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Definition

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.


Also see


Sources