Minimum of a coercive convex functional on a Hilbert space (${\min }_{x\in K}F(x)$)

Let $K$ be a nonempty norm-closed convex set in a real Hilbert space, and let $F:K\to \mathbb{R}$ be continuous and a convex function. If $K$ is bounded, or $F(x)\to \infty$ as $x\in K$ tends to infinity in norm, then $F$ attains its infimum. Every sublevel set is bounded, norm-closed, and convex, hence contains the weak limits of its sequences by norm-closed convex set is weakly closed. A minimizing sequence in one fixed sublevel set has a weakly convergent subsequence by weak sequential compactness in a Hilbert space. Its limit lies in every sublevel set above the infimum. Since $F$ is real-valued, this also rules out infimum $-\infty$ and proves attainment. This is an instance of the direct method in the calculus of variations.