Definition:C^k Norm

Definition
Let $f \in \map {C^k} \R$ 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 $x$
 * $\norm {\, \cdot \,}_\infty$ denotes the supremum norm.

Also denoted as

 * $\norm {\, \cdot \,}_{\map {C^k} \R}$

can be seen written as


 * $\norm {\, \cdot \,}_{k, \infty}$