From the Preface: ""This textbook has evolved from a set of lecture notes...In both the course and the book, I have in mind first- or second-year graduate students in Mathematics and related fields such as Physics...It is necessary for the reader to have a foundation in advanced calculus which includes familiarity with: least upper bound (LUB) and greatest lower bound (GLB), the concept of function, $\epsilon$'s and their companion $\delta$'s, and basic properties of sequences of real and complex numbers (convergence, Cauchy's criterion, the Weierstrass-Bolzano theorem). It is not presupposed that the reader is acquainted with vector spaces..., matrices..., or determinants...There are over four hundred exercises, most of them easy...It is my hope that this book, aside from being an exposition of certain basic material on Hilbert space, may also serve as an introduction to other areas of functional analysis.
Vector Spaces: 1. Complex vector spaces 2. First properties of vector spaces 3. Finite sums of vectors 4. Linear combinations of vectors 5. Linear subspaces, linear dependence 6. Linear independence 7. Basis, dimension 8. Coda Hilbert Spaces: 1. Pre-Hilbert spaces 2. First properties of pre-Hilbert spaces 3. The norm of a vector 4. Metric spaces 5. Metric notions in pre-Hilbert space; Hilbert spaces 6. Orthogonal vectors, orthonormal vectors 7. Infinite sums in Hilbert space 8. Total sets, separable Hilbert spaces, orthonormal bases 9. Isomorphic Hilbert spaces; classical Hilbert space Closed Linear Subspaces: 1. Some notations from set theory 2. Annihilators 3. Closed linear subspaces 4. Complete linear subspaces 5. Convex sets, minimizing vector 6. Orthogonal complement 7. Mappings 8. Projection Continuous Linear Mappings: 1. Linear mappings 2. Isomorphic vector spaces 3. The vector space 4. Composition of mappings 5. The algebra 6. Continuous mappings 7. Normed spaces, Banach spaces, continuous linear mappings 8. The normed space 9. The normed algebra, Banach algebras 10. The dual space Continuous Linear Forms in Hilbert Space: 1. Riesz-Frechet theorem 2. Completion 3. Bilinear mappings 4. Bounded bilinear mappings 5. Sesquilinear mappings 6. Bounded sesquilinear mappings 7. Bounded sesquilinear forms in Hilbert space 8. Adjoints Operators in Hilbert Space: 1. Manifesto 2. Preliminaries 3. An example 4. Isometric operators 5. Unitary operators 6. Self-adjoint operators 7. Projection operators 8. Normal operators 9. Invariant and reducing subspaces Proper Values: 1. Proper vectors, proper values 2. Proper subspaces 3. Approximate proper values Completely Continuous Operators: 1. Completely continuous operators 2. An example 3. Proper values of CC-operators 4. Spectral theorem for a normal CC-operator Appendix Index.