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I am now studying Bayesian inverse problems. In the note of Dashti and Stuart https://arxiv.org/abs/1302.6989, they mentioned that "... when considering a non-separable Banach space $B$, it is ...

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 ...