Q:

infinite dimensional polyhedra

I have a reference request which I hope some reader here can help me with.
I have encountered a set that has all the properties that one would expect from a polyhedral set (in the sense of finite dimensional convex analysis - an intersection of finitely many half-spaces), however in my case the number of intersecting half-spaces could be infinite. I am interested in things such as extreme points, extreme rays etc. However, the catch is that the set is itself in the space of functions. I am therefore looking for a principled generalization of the concept related to polyhedra to infinite-dimensional spaces. Could someone help me out with this?

A:

Intersecting countably many half-spaces, you can get a general closed convex set (in a separable Banach space, say).  For extreme points and other interesting things, I like:
Robert R. Phelps, Lectures on Choquet's Theorem, second edition, Lecture Notes in Mathematics, Vol. 1767. Springer- Verlag, 2001.

