The conditional expectation as estimator of normally distributed random variables with values in infinitely dimensional Banach spaces

Given the linear model b = Ax - [var epsilon], where x and [var epsilon] are Gauss distributed with covariance operators Rx and R[var epsilon], R[var epsilon] positive definite; then b(x) = RxA'(ARxA' + R[var epsilon])-1b is the expectation of the conditional distribution of x relative to b. This is well known for finite dimensions. This formula is generalized for x [epsilon] E and [var epsilon] [epsilon] F, where E is a real, separable Banach space and F is a real, normed vector space.