# Reference on Probability theory on functional spaces (in special Hilbert spaces)

Currently, I am working on some sort of stochastic optimization problems defined over function spaces.

I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples"). However, the standard discussions are restricted to random objects in Euclidean spaces (random vectors and variables). I like to know whether there are some proper references on extensions of these notions and theorems (WLLN, SLLN, CLT, Conditional Expectation,...) to the case where the objects of interest belong to a function space.

Thanks