Courses

Hilbert spaces olinear operators on Hilbert spaces. We start with the introduction of the scalar product spaces. Then we discuss the metric induced by the scalar product and start the study of the Hilbert spaces. Next, we investigate the ortho-normal systems in scalar product spaces. We study the bounded operators and functionals. We prove the Riesz rep- resentation theorem for the bounded functionals on Hilbert spaces. Then we give the definition of the adjoint operator for the bounded operators and prove its basic properties. We finish the chapter with the discussion of the decomposition of spectrum of bounded operators. We deal with the compact operators. We work with the closed operators and prove the so-called "closed graph theorem". We introduce the adjoint operator already for unbounded operators and construct the adjoints for some di erential operators. We study the spectrum of unbounded operators and prove the "Spectral Theorem". We finish the course with the theory of convergence of operators: norm resolvent, strong resolvent and weak resolvent convergence.