Recall that the Krein-Milman theorem asserts that a compact convex set in a LCTVS is the closed convex hull of its extreme points. This has lots of applications to areas of mathematics that use analysis: the existence of pure states in C*-algebra theory, the existence of irreducible representations of groups, the existence of ergodic measures…
I'm interested in applications of the theorem which are very easy to state but hard to achieve any other way. When I say "very easy to state" I mean the result should be expressible in the language of elementary Banach space or Hilbert space theory – no C*-algebras, representation theory, or measures. For an example of what I have in mind, the Krein-Milman theorem implies that $C[0,1]$ is not the dual of any Banach space. If anyone knows an application of Krein-Milman to the theory of Fourier series, that would be ideal.
Edit: (t.b.) A version of this question now is on MathOverflow and already has a few answers.
Best Answer
The Krein-Milman theorem is one way to prove De Finnetti's theorem: that every exchangeable sequence of random variables can be seen as a random draw among i.i.d. random variables.
The proof still involves the nontrivial step of showing that the i.i.d. distributions are the extreme points of that set, so it may not be as elementary as you want. It can also be proven in other ways; what's harder is ultimately subjective.
Nevertheless, I think it's a great way to illustrate the power of the theorem, because the statement itself is very easy to understand, and the result is surprising.