# Normal correlation theorem in Banach spaces

I am looking for a version of the Bayesian Linear Model in Banach spaces.

## Background: the finite dimensional case

The following is a well-known theorem about normal correlation for finite-dimensional random vectors.

Suppose that $X$ and $Y$ are random vectors with means $\mu_X$, $\mu_Y$, variance matrices $\Lambda_X$, $\Lambda_Y$ and cross-covariance $\Lambda_{XY}$. If $X$ and $Y$ are jointly Normally distributed, then the marginal distribution of $X$ is Normal and the conditional distribution of $Y$ given $X$ is Normal with mean $$\mu_Y + \Lambda_{YX}\Lambda_X^{-1}(X-\mu_X)$$ and variance $$\Lambda_Y - \Lambda_{YX}\Lambda_X^{-1}\Lambda_{XY}.$$ Conversely, if the distribution of $X$ is normal and the conditional distribution of $Y$ given $X$ is Normal with mean $AX$ (for some matrix $A$) and constant variance $\Lambda_{Y|X}$, then the joint distribution of $X$ and $Y$ is Normal.

From the previous result one can derive the following corollary.

Suppose that $$X\sim N(\mu_X, \Lambda_X)$$ and $$Y|X\sim N(AX, V).$$ Then $$X|Y\sim N(\mu_{X|Y}, \Lambda_{X|Y}),$$ where $$\mu_{X|Y} = \mu_X + K(Y - A\mu_X),$$ $$\Lambda_{X|Y} = \Lambda_X - K(A\Lambda_XA' + V)K',$$ with $K = \Lambda_XA'(A\Lambda_XA' + V)^{-1}$.

In all the above formulas, a generalized inverse can replace the inverse in case the inverse does not exist.

## The infinite dimensional case

This is actually what I am looking for. More specifically,

Is anybody aware of a similar result for Banach space-valued random vectors? If not, what are the closest results that I can look at?

From a statistical point of view, the result above provides an updating formula for the Normal-Normal linear model.

Is there a Bayesian version of the linear model for Banach space-valued random vectors?

In all of my questions, I would be happy with results or pointers for separable Banach spaces and also for the simpler case of Hilbert space-valued random variables.

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