Definition of isonormal Gaussian process

functional-analysismalliavin-calculusprobability distributionsstochastic-processes

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

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

Best Answer

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.

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