Hilbert Spaces

Abstract

Abstract This chapter is a good introduction to Hilbert spaces and the elements of operator theory. The two leading sections contains staple topics such as the projection theorem, projection operators, the Riesz representation theorem, Bessel’s inequality, and the characterization of separable Hilbert spaces. Sections 7.3 and 7.4 contain a rather detailed study of self-adjoint and compact operators. Among the highlights are the Fredholm theory and the spectral theorems for compact self-adjoint and normal operators, with applications to integral equations. The section exercises contain problems that suggest alternative approaches, thus allowing the instructor to shorten the chapter while preserving good depth. The last section extends the results to compact operators on Banach spaces. The chapter contains more results than is typically found in an introductory course.