I'm sure I'll find the answer in Rudin's Functional Analysis, but perhaps a helpful reader will educate me and the rest; otherwise, what's the point of blogging?
Say we have a Hilbert space H -- so it's locally convex and the Krein-Milman theorem applies. Let K be a compact convex set in H. Is it still true that linear functionals achieve their maxima on extreme points of K? What about convex functionals? [I should really look up the latter in the last chapter of Borwein and Lewis.]
Seems like these should be true, but I've learned to mistrust my intuition in high (and a forteriori infinite) dimensions. Anyone have an answer handy?