Hilbert Spaces

Abstract

The problems in this chapter deal with Hilbert spaces and linear operators. Section 1.1 emphasizes the notions of dense subspaces, of complementary subspaces, and the fundamental concept of complete system. Many exercises are then proposed on the Fourier expansion, in the context of “abstract” Hilbert spaces, in the space of sequences $\ell ^2$, and in the “concrete” space of square-integrable functions $L^2$, with several examples and applications, which include some typical Dirichlet and Neumann Problems. Section 1.2 is devoted to studying the different properties of linear operators between Hilbert spaces: their domains, ranges, norms, boundedness, closedness, and to examining special classes of operators: adjoint and self-adjoint operators, projections, isometric and unitary operators, functionals, and time-evolution operators. Great attention is paid to the notion of eigenvalues and eigenvectors. Many exercises propose the different procedures needed for finding eigenvectors and the extremely various situations which can occur. Another frequent question concerns the different notions of convergence of given sequences of operators.