Let $X$ be a reflexive [Banach space](/page/Banach%20Space), let $Y$ be a Banach space, and let $j:X\to Y$ be a continuous linear embedding with the following compactness property: for every bounded sequence $(v_k)_{k=1}^{\infty}$ in $X$, the sequence $(jv_k)_{k=1}^{\infty}$ has a norm-convergent subsequence in $Y$. Let $I_0:X\to(-\infty,\infty]$ be sequentially weakly lower semicontinuous on $X$, and let $J:Y\to\mathbb R$ be continuous with respect to the norm topology on $Y$. Define $I:X\to(-\infty,\infty]$ by $I[u]=I_0[u]+J(ju)$. Then $I$ is sequentially weakly lower semicontinuous on bounded subsets of $X$: whenever $(u_k)_{k=1}^{\infty}$ is a bounded sequence in $X$ and $u_k\rightharpoonup u$ in $X$, one has

\begin{align*} I[u]\le \liminf_{k\to\infty} I[u_k]. \end{align*}