I have been looking to generalize a result concerning random variables with values in $\mathbb{R}^d$ to random variable with values in function spaces (in particular a space of smooth functions). I noticed that expressing many of my conditions in the $\mathbb{R}^d$-case as norms and inner-products made it very easy to translate my results into the functional case. It seemed very well-behaved, so I pondered if maybe further generalization was possible.
Does anyone have a reference about probability theory on Hilbert spaces in general, i.e. how to define/calculate expectations, variances, covariances of Hilbert space valued random variables? I'm very familiar with measure theory and probability theory but not very solid on heavy functional analysis / topology.
I'd like to avoid thinking of stochastic processes as indexed families of regular random variables but instead view them as points in some function space (Hilbert space) and then use the tools of the function space to do probability. Maybe this isn't possible and it is just wishful thinking from me but a solid book or two would be much appreciated!