= Solution
The <Sobolev space> $H^1(U)$ is a separable <Hilbert space>. A bounded sequence therefore has a weakly convergent subsequence by the <weak subsequence of a bounded Hilbert-space sequence>; write $u_{i_j}\rightharpoonup u$ in $H^1(U)$. Since $U$ is bounded with smooth boundary, the <Rellich-Kondrachov compactness theorem> says that $H^1(U)\hookrightarrow L^2(U)$ is compact. Passing to a further subsequence gives
$$
\boxed{u_{i_j}\longrightarrow u\quad\text{strongly in }L^2(U).}
$$
Back to article page
