relationship between the Hilbert space of random variables and computational least squares

Question

It is clear to me that in the Hilbert space $\mathcal{H}$ of random variables the estimate of X is the projection of X onto the subspace spanned by the predictor(s) Y, i.e. the conditional expectation of X given Y. It is also common practice in n-dimensional case space to project the observed sample of realizations of X onto the model manifold, say $Y = Y (\theta)$. I cannot see how the two situations are linked. Indeed, suppose $X : \Omega \rightarrow \mathbb{R}^n$. Then whereas obviously $X : \omega \mapsto \boldsymbol{r}$, also $\omega : X \mapsto \boldsymbol{r}$. So we could think of the realization as the map between the space of random variables and their observations, $\omega : \mathcal{H} \rightarrow \mathbb{R}^n$. It is not clear to me (at all) that such a map would preserve projections or other relevant properties. Any words of wisdom or clear references?