Let $X$ be a reflexive [Banach space](/page/Banach%20Space), let $Y$ be a Banach space, and let

\begin{align*} j:X\to Y \end{align*}

be a continuous injective linear compact embedding, denoted $X\hookrightarrow\hookrightarrow Y$. Identify each $u\in X$ with its image $j(u)\in Y$ when discussing convergence in $Y$. Let $\mathcal A\subset X$ be sequentially weakly closed in $X$. If $(u_k)_{k=1}^{\infty}$ is a bounded sequence in $\mathcal A$, then there exist $u\in\mathcal A$ and a subsequence $(u_{k_j})_{j=1}^{\infty}$ such that

\begin{align*} u_{k_j}\rightharpoonup u \quad \text{in } X \end{align*}

and

\begin{align*} u_{k_j}\to u \quad \text{in } Y. \end{align*}