Metric spaces, normed vector spaces, basis sets. The four subspaces of the linear transforms.  Approximation in Hilbert spaces: least squares filtering and estimation, linear regression, polynomial approximation, minimum norm solutions and system identification. Matrix factorization, .eigenvectors, singular value decomposition, iterative matrix inverses, pseudoinverse. Theory of constrained optimization and dynamic programming. Expectation – maximization algorithm. Kalman filtering.