For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this:

Let $H(\Omega;\mathbb{R})$ be a seperable Hilbert space. A random variable $u \in H$ is said to be a Gaussian random variable if $\langle u,v \rangle$ is Gaussian (i.e., $\langle u,v \rangle$ follows the standard Normal distribution on $\mathbb{R}$) for all $v \in H$.

Even more generally, I found in this lecture note (Definition 2.1) that Gaussian r.v. in topological vector space is defined as follows

Let $W$ be a topological vector space, and $\mu$ a Borel probability measure on $W$. $\mu$ is Gaussian iff, for each continuous linear functional $f\in W_*$, the pushforward $\mu \circ f^{−1}$ is a Gaussian measure on $\mathbb{R}$.

My question might be a bit vague: I would like to know the intuition and motivation of such definition. It seems to me that the infinite-dimensional Gaussian measure is defined via its finite-dimensional analogy. I can think of the analogy from characteristic function $\mathbb{E}[e^{i \langle u,v \rangle}]$ or that linear combinatins of Gaussians <=> Gaussian. What benifits does this kind of definition bing? Are there other definitions of Gaussian r.v.s in infinite-dimensional spaces?

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