The authors develop a self-contained theory for linear estimation in Krein spaces, The derivation is based on simple concepts such as projections and matrix factorizations and leads to an interesting connection between Krein space projection and the recursive computation of the stationary points of certain second-order (or quadratic) forms, The authors use the innovations process to obtain a general recursive linear estimation algorithm, When specialized to a state-space structure, the algorithm yields a Krein space generalization of the celebrated Kalman filter with applications in several areas such as H-infinity-filtering and control, game problems, risk sensitive control, and adaptive filtering.