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If $f \in C^k ( \mathbb R)$ and $A \subset \mathbb R$, what is the usual meaning of the notation below? $ \| f \|_{C^k (A)} $

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Assuming you meant $A\subseteq\mathbb{R}$, the notation usually means $\|f\|_{C^k(A)}=\|f\|+\|f'\|+\cdots+\|f^{(k)}\|$ where $f^{(i)}$ is the $i$th derivative of $f$, and where for a function $g:A\to\mathbb{R}$, the notation $\|g\|$ refers to the sup norm: $\|g\|=\sup_{x\in A} \,|g(x)|$ I've also seen the definition $\|f\|_{C^k(A)}=\max\left\{\|f\|,\|f'\|,\ldots,\|f^{(k)}\|\right\}$ which (if I remember correctly) induces the same topology.

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    As a remark, both norms you mention do in fact induce the same topology because the first norm is $\|(\|f\|, \|f'\|, ..., \|f^{(k)}\|)\|_\infty$, the second is $\|(\|f\|, \|f'\|, ..., \|f^{(k)}\|)\|_1$ and all norms on a finite dimensional space are equivalent.2012-07-01