Let $X$ be a reflexive [Banach space](/page/Banach%20Space), let $K \subset X$ be nonempty, norm-closed, and convex, and let $I:K\to(-\infty,\infty]$ be sequentially weakly lower semicontinuous. Assume that $I$ is coercive on $K$, meaning that for every sequence $(u_j)_{j=1}^{\infty}$ in $K$ with $\|u_j\|_X\to\infty$, one has $I[u_j]\to\infty$. If

\begin{align*} \inf_{v\in K} I[v] < \infty, \end{align*}

then there exists $u_*\in K$ such that

\begin{align*} I[u_*]=\inf_{v\in K} I[v]. \end{align*}