Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite in $H^1(B)$, where $B$ is an open ball ? I have tried using the fallowing result from Evans, 2010, page 268:
Theorem: Assume $U$ is bounded and $\partial U$ is $C^1$. Select a bounded open set $V$ such that $U \subset \subset V$. Then there exists a bounded linear operator $$ E : H^1(U) \rightarrow H^1(\mathbb{R}^N) $$ such that for each $u \in H^1(U)$: i) E(u) = u, a.e. in $U$, ii) Eu has support within $V$, and iii) Exists a constant $C > 0$ such that $$ ||E(u)||_{H^1(\mathbb{R}^N)} \leq C ||u||_{H^1(U)}. $$