Monday, February 19, 2007

Linear/convex programming in infinite dimensions?

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?

1 comment:

Anand said...

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.