What mathematical background do I need for the Gaussian Process book by Rasmussen and Williams? I started reading the book today, but right off the bat, he mentions infinite Hilbert spaces in the notation, so I feel that it might be out of my league. I am familiar with linear algebra, probability and statistics, and basic multivariable calculus (up to Green's theorem).
 A: I'd say skip whatever occasional math that is not central to understanding GP. It is more important to understand the regression problem first without all that Bayesian framework.  If you have worked on a regression problem and tuned the (hyper)parameters using cross validation then going to GP you will appreciate the Bayesian setting more and understand it more clearly.  
But there are some math not to skip and working  with the gpml software by Rasmussen et al. on a toy problem greatly helps to figure that out.  Sometimes your intuition does not match with what you are observing and you go back to the math to find out why and you realize you misunderstood.  Reading the book and experimenting with gpml go hand in hand.
A: From my experience - linear algebra, calculus and basic probability are all that are required initially. The book is very well written and does a good job in explaining GP's without expecting the reader to have much previous knowledge.
Obviously there are strong links between GPML and other areas of machine learning, so if you have knowledge of SVMs or any of the regression based ML methods they will stand to your advantage - although this definitely isn't required.
If you really want to understand GPs make sure you write your own as you read through the book; that was invaluable to me. Also when you get to the chapter on kernels you should check out the Kernel Cookbook which is a great little resource.
