How could the joint normal distribution be available from multivariate normal distribution and linearity?

Question

The picture below is Definition 1.1.1 and two Remarks in page 4 of Book Nualart: The Malliavin Calculus and Related Topics (2ed).

I was able to understand the definition, it is the "Gaussian process" indexed by a Hilbert space. But I cannot figure out the second remark, as asked in the title, how could the joint normal distribution be available from multivariate normal distribution and linearity?

Any comments or hints will be appreciated.

Answer 1

By a Gaussian family, it means that, for any $h_1, \ldots, h_n$, $W(h_1), \ldots, W(h_n)$ are jointly normal. That is, for any real numbers $a_1, \ldots, a_n$, $$a_1 W(h_1) + \cdots + a_n W(h_n)$$ is normal, which is obviously true, as, from the linearity, \begin{align*} a_1 W(h_1) + \cdots + a_n W(h_n) = W(a_1h_1 + \cdots + a_n h_n), \end{align*} which is normal by definition.