\begin{definition}[Definition:C^k Norm]
Let $f \in \map {\CC^k} \R$ be a function of differentiability class $k$.
Then '''$C^k$ norm''' is defined as:
:$\ds \norm f_{\map {C^k} \R} := \sum_{i \mathop = 0}^k \sup_{x \mathop \in \R} \size {\map {f^{\paren i} } x} = \sum_{i \mathop = 0}^k \norm {f^{\paren i} }_\infty$
where:
:$f^{\paren i}$ denotes the $i$-th derivative {{WRT|Differentiation}} $x$
:$\norm {\, \cdot \,}_\infty$ denotes the supremum norm.
\end{definition}