Do extracted weak $H^{1,2}$-limits and $C^0$-limits coincide? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:57:43Z http://mathoverflow.net/feeds/question/65337 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65337/do-extracted-weak-h1-2-limits-and-c0-limits-coincide Do extracted weak $H^{1,2}$-limits and $C^0$-limits coincide? Orbicular 2011-05-18T14:51:03Z 2011-05-18T15:00:19Z <p>Let $I$ be a bounded interval and consider a sequence $(u_k)$ in $H^{1,2}(I)$ (usual Sobolev space). Suppose furthermore, that the sequence $(u_k)$ is bounded in $H^{1,2}(I)$. Then, by Rellich, we can extract a subsequence, still denoted by $u_k,$ s.t. $u_k$ converges to some $\bar{u}$ in $C^0(I)$. Furthermore, by weak compactness of bounded sets in $H^{1,2}(I)$ we can select a subsequence, s.t. $u_k$ converges to some $u$ weakly in $H^{1,2}(I)$. Thus, $u_k$ converges to $\bar{u}$ in $C^0$ and weakly to $u$ in $H^{1,2}$.</p> <p>Do these limits coincide, i.e. is it true that $u=\bar{u}$?</p> http://mathoverflow.net/questions/65337/do-extracted-weak-h1-2-limits-and-c0-limits-coincide/65339#65339 Answer by Michael Renardy for Do extracted weak $H^{1,2}$-limits and $C^0$-limits coincide? Michael Renardy 2011-05-18T15:00:19Z 2011-05-18T15:00:19Z <p>Either type of convergence implies distributional convergence, among other things. So the limits must be the same.</p>