What mean that expression "closed respect a norm". For example:
$H^{m,p}(\Omega)= \overline{\left\{{u\in C^m(\Omega); \|u\|_{W^{m,p}}<\infty}\right\}}^{\|\cdot\|}$
What mean that expression "closed respect a norm". For example:
$H^{m,p}(\Omega)= \overline{\left\{{u\in C^m(\Omega); \|u\|_{W^{m,p}}<\infty}\right\}}^{\|\cdot\|}$
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.
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.