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 fortiori*infinite) dimensions. Anyone have an answer handy?

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I think the answer is also in Royden, but that's at work and I'm at home, where little work gets done and I keep few math books.

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