Diego Elements of Hilbert Space and Operator Theory with Application to Integral Equations

Preface These lecture notes present elements of Hilbert space and the theory of linear operators on Hilbert space, and their application to integral equations. Chapter 1 reviews vector spaces: bases, subspaces, and linear transforms. Chapter 2 covers the basics of Hilbert space. It includes the concept of Ba-nach space and Hilbert space, orthogonality, complete orthogonal bases, and Riesz representation. Chapter 3 presents the core theory of linear operators on Hilbert space, in particular, the spectral theory of linear compact operators and Fredholm Alternatives. Chapter 4 is the application to integral equations. Some supplemental materials are collected in Appendicies. The prerequisite for these lecture notes is calculus. Occasionally, advanced knowledges such as the Lebesgue measure and Lebesgue integration are used in the notes. The lack of such knowledge, though, will not prevent readers from grasping the essence of the main subjects. The presentation in these notes is kept as simple, concise, and self-complete as possible. Many examples are given, some of them with great details. Exercise problems for practice and references for further reading are included in each chapter. The symbol ⊓ ⊔ is used to indicate the end of a proof. These lecture notes can be used as a text or supplemental material for various kinds of undergraduate or beginning graduate courses for students of mathematics, science, and engineering.