Let $\mathcal{I}$ be the set of increasing uniformly bounded real functions on the closed unit interval, equipped with the inner product $\langle f,g\rangle=\int_0^1 f(x)g(x) dx$.
Let $F:\mathcal{I}\to[0,1]$ be a continuous convex functional and $\mathcal{J}\subseteq \mathcal {I}$ be a convex and compact subset. Show that there is (a separating hyperplane) $g\in \mathcal{I}$ and $\gamma\in \mathbb{R}$ such that $\langle g,x\rangle \leq\gamma$ for all $x\in \mathcal{J}$, and if the inequality holds with equality, then $x$ a maximizer of $F$ on $\mathcal{J}$.