Definition of isonormal Gaussian process

Question

The definition of an isonormal Gaussian process (from Nualart's book: The Malliavin Calculus and related topics) is as follows:

My question is: why we want the space $H$ to be a real separable Hilbert space?

Answer 1

If you read the rest of the first chapter, there is a paragraph entitled "Wiener's Chaos Decomposition". This Decomposition is only valid if the surrounding space is separable . I don't have the book with me.

Answer 2

I think the real restriction is not that essential. It's just simpler. One does not have complex conjugates all over the place in the definitions. It is also related to the desire to model real-valued processes like standard Brownian motion. The separable hypothesis is more important. That makes $H$ a Polish space which is nice for probability. Note however that here the space that really needs to Polish is typically a Banach space that is bigger than $H$. See the notion of abstract Wiener space.