Normed vector spaces, Banach/Hilbert spaces. Linear operators, adjoint and invers operator. Linear systems as operators, adjoint systems, stability. Quadratic optimization-problems for linear systems. Causal and time-invariant systems, Hankel/Toeplitz operators, transfer function.
Topological vector spaces, linear functionals, dual space. Weak topologies. Optimization in Banach/Hilbert spaces. Min-max theorem and duality. Minimum norm theorems. Nehari theorem and other extremal problems in Hardy spaces. Hahn-Banach theorem and separation of convex sets. Convex analysis in normed spaces.