Let $X$ be a reflexive [Banach space](/page/Banach%20Space), let $\mathcal A \subset X$ be nonempty and sequentially weakly closed, and let

\begin{align*} I:\mathcal A \to \mathbb R \cup \{+\infty\} \end{align*}

be bounded below, coercive on $\mathcal A$, and sequentially weakly lower semicontinuous on $\mathcal A$. Here $\mathbb N=\{1,2,3,\dots\}$, coercive on $\mathcal A$ means that for every sequence $(u_k)_{k=1}^{\infty}$ in $\mathcal A$ with $\|u_k\|_X \to \infty$, one has $I[u_k]\to +\infty$, and sequential weak lower semicontinuity on $\mathcal A$ means that whenever $(u_k)_{k=1}^{\infty}$ is a sequence in $\mathcal A$ and $u_k\rightharpoonup u$ weakly in $X$ for some $u\in\mathcal A$, then

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

Then there exists $u_*\in\mathcal A$ such that

\begin{align*} I[u_*]=\inf_{u\in\mathcal A} I[u]. \end{align*}