Definition:Supremum Metric/Differentiability Class

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

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

Let $\mathscr D^r \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$ which are of differentiability class $r$.

Let $d: \mathscr D^r \closedint a b \times \mathscr D^r \closedint a b \to \R$ be the function defined as:
 * $\ds \forall f, g \in \mathscr D^r \closedint a b: \map d {f, g} := \sup_{\substack {x \mathop \in \closedint a b \\ i \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x}$

where:
 * $f^{\paren i}$ denotes the $i$th derivative of $f$
 * $f^{\paren 0}$ denotes $f$
 * $\sup$ denotes the supremum.

$d$ is known as the supremum metric on $\mathscr D^r \closedint a b$.

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

Also see

 * Supremum Metric on Differentiability Class is Metric