Let $(\Omega, \mathcal{F}, P)$ be a probability space. We define a random variable on a Hilbert space $H$ as a measurable function $$ X : \Omega \to H $$ where $H$ is equipped with its associated Borel $\sigma$-algebra. We then say that $X$ is a Gaussian element with mean $\mu \in H$ and covariance operator $\mathcal{C}:H \to H$ if $$ \langle X,f \rangle\sim \mathcal{N}(\langle \mu, f \rangle,\langle \mathcal{C}f,f \rangle) $$ for any $f\in H$. On the wikipedia page for the Gaussian distribution it is stated that a Gaussian process can be viewed as a Gaussian element in some Hilbert space $H$. Now, the usual definition of a Gaussian process is given by stating that its finite dimensional projections are all multivariate gaussians - it does not specify any Hilbert spaces as above.
I am wondering how the Gaussian process can be viewed as an element of a Hilbert space. In particular, which functions do we include in the Hilbert space? Any help/sources would be appreciated.
Link to the wiki-article: https://en.wikipedia.org/wiki/Normal_distribution#Related_distributions