The linear optimal filtering problems in infinite dimensional Hilbert spaces and their extensions are investigated. The quality functional is allowed to be a general quadratic functional defined by a possibly degenerate operator. We describe the solution of the stable and the causal filtering problems. In the case of causal filtering, we establish the relation with a relaxed causal filtering problem in the extended space. We solve the last problem in continuous and discrete cases and give the necessary and sufficient conditions for the solvability of the original causal problem as well as the conditions for the analogue of the Bode-Shannon formula to de ne an optimal filter.