If $f \in C^k ( \mathbb R)$ and $A \subset \mathbb R$, what is the usual meaning of the notation below? $$ \| f \|_{C^k (A)} $$
What is the usual meaning of the notation $ \| f \|_{C^k (A)}$?
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1 Answers
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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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0Oh thank you it was just a sup. – 2012-07-01
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0Well, it's the sum of the sup-norms of the function itself, its first derivative, second derivative, etc., and $k$th derivative. – 2012-07-01
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0I got it, thank you very much. – 2012-07-01
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0As 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