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What mean that expression "closed respect a norm". For example:

$H^{m,p}(\Omega)= \overline{\left\{{u\in C^m(\Omega); \|u\|_{W^{m,p}}<\infty}\right\}}^{\|\cdot\|}$

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    The grammatical version of "closed respect a norm $\|\cdot\|$" is *closed with respect to the norm* $\|\cdot\|$"2012-11-07

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It means that in the topology generated by this norm, the set is closed. Since norms are metrics, this means that all the $\|\cdot\|$-convergent sequences from this set have limits inside this set as well.

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    @user46060: I just saw now that those are different norms. That doesn't matter, though. This is just a definition of a particular set, and you claim that it is closed under $NORM_2$, meaning all $NORM_2$-convergent sequences find their limit within this set.2012-11-07
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It means that you take the closure of $\left\{{u\in C^m(\Omega); \|u\|_{W^{m,p}}<\infty}\right\}$ with respect to $\| \cdot \|$. The closure is the set itself together with all its limit points. And a limit point is a point such that every $\varepsilon$-ball around it has non-empty intersection with the set.