I am reading the very first chapter of "Gaussian Hilbert Spaces" by S. Janson.
Definition: A Gaussian Hilbert space is a closed subspace of $L^2(\Omega, \mathcal{F}, P)$ consisting of centered Gaussian random variables.
Example 1: Let $\xi$ be any non-degenerate, normal variable with mean zero. Then $\{ t \xi: t \in \mathbf{R} \}$ is a one-dimensional Gaussian Hilbert space.
Example 2: Let $\xi_1, \dots, \xi_n$ have a joint normal distribution with mean zero. Then their linear span $\{ \sum_{i = 1}^n t_i \xi_i : t_i \in \mathbf{R} \}$ is a finite-dimensional Gaussian Hilbert space.
I am missing something really fundamental and, presumably, trivial for other people. In both cases, the spaces are composed of one-dimensional Gaussian random variables with zero means and all possible finite variances. Then the question is: what Gaussian random variable cannot be obtained using the first approach while it can easily be constructed using the second one? What is the difference? Or may be it is not about individual variables but rather about their relationships with each other. In the first case, all the variables are perfectly correlated whereas the second space is much reacher in this regard, especially, if we assume that $\xi_1, \dots, \xi_n$ are independent.
Thank you!
Regards, Ivan
The difference lies in the dimension of the Hilbert space generated. Take $\{ \xi_1, \xi_2\} \sim N(0, I_2)$, which form an orthonormal basis of a two dimensional subspace. This is clearly not the same as the one dimensional subspace generated by $\xi_1$.
Indeed you seem to be missing something fundamental: the distribution of a random variable $\xi$, or equivalently, its push-forward measure on $\mathbb{R}$, does not characterize $\xi$. For example, if $\xi \sim U[0,1]$, then $ -\xi +1 \sim U[0,1]$.