Reference: Gaussianity of linear functional of Gaussian process

Question

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets,

If you take $X$ in $C([a,b])$ [i.e., $X$ is a Gaussian process with sample paths in the space of continuous functions on the interval $[a,b]$], then $b^*(X)$ is Gaussian on the real line for any $b^*$ in the dual of $C([a,b])$, which is the space of [integrals against] measures of bounded variation.

"Continuity" is with respect to, e.g., the "intrinsic semimetric" in van der Vaart and Wellner (1996, e.g. p. 41). Of course, it's fine to have a result for a general Banach space and its dual, where this is just a special case.

I've been told to look in Bogachev's "Gaussian Measures" (which I have from the library), but I have not spotted such a result; either the book is too dense, or I am too dense, or both. (Maybe Definition 2.2.1(ii), p. 42?)

Answer 1

You have the correct definition in Bogachev's book: to say that $X$ "is Gaussian with values in $C([a,b])$" is to say that $X$ is a random element of $C([a,b])$ with the property that $b^∗(X)$ is normally distributed for each $b^∗$ in the dual of $C([a,b])$.