Definition:C^k Norm

Definition

Let $f \in \map {C^k} \R$ be a function of differentiability class $k$.

Then the $C^k$ norm of $f$ 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 with respect to $x$

$\norm {\, \cdot \,}_\infty$ denotes the supremum norm.

Also denoted as

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

can be seen written as

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

Also see

• $C^k$ Norm is Norm

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces