  In this paper we investigate the conditional distributions of two Banach
space valued, jointly Gaussian random variables. We show that these conditional
distributions are again Gaussian and that their means and covariances are
determined by a general finite dimensional approximation scheme based upon a
martingale approach. In particular, it turns out that the covariance operators
occurring in this scheme converge with respect to the nuclear norm and that the
conditional probabilities converge weakly. Moreover, we discuss in detail, how
our approximation scheme can be implemented in several classes of important
Banach spaces such as (reproducing kernel) Hilbert spaces and spaces of
continuous functions. As an example, we then apply our general results to the
case of Gaussian processes with continuous paths conditioned to partial but
infinite observations of their paths. Here we show that conditioning on
sufficiently rich, increasing sets of finitely many observations leads to
consistent approximations, that is, both the mean and covariance functions
converge uniformly and the conditional probabilities converge weakly. Moreover,
we discuss how these results improve our understanding of the popular Gaussian
processes for machine learning.
