  These notes started to educate ourselves and to collect some background for
our future work, with the hope that perhaps they will be useful to others also.
Many if not all results are more or less elementary or available in the
literature, but we need to fill some holes (which are undoubtely statements so
trivial that the authors we use do not consider them holes at all) or make
straightforward extensions, and then we do the proofs in sufficient detail for
reference. Topics include random fields and stochastic processes as random
elements in Hilbert spaces, Karhunen-Lo\`{e}ve explansion and random
orthonormal series, laws of large numbers, white noise, convergence of the
Ensemble Kalman Filter (EnKF), and the ensemble Kalman Transform Filter (ETKF).
