I asked this over on cross validated, but thought it might also get an answer here:
The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the separable Hilbert spaced valued case, i.e., for $(X,Y)$, $$ \mu_{X|Y=y} = \mu_X - C_{XY}C_{Y}^{-1}(\mu_Y - y) $$ and $$ C_{X|Y=y} = C_{X} - C_{XY}C_Y^{-1}C_{YX} $$
I was trying to trace a proof for this in the separable Hilbert space case, and all the papers I found tended to point to Linear Estimators and Measurable Linear Transformations on a Hilbert Space by A. Mandelbaum (1984). Digging through that paper, there's one a part of the proof I'm stumbling on:
I'm struggling with that first equality in (3.6), where the conditional expectation becomes the summation. Thanks for any help. Alternatively, if someone has a reference to another (better?) proof, let me know.