This book provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential equations, integral equations, infinite systems of linear equations, approximation theory, and numerical analysis. Designed as a textbook for senior undergraduate and graduate students, it begins with the geometry of Hilbert spaces and proceeds to the theory of linear operators on these spaces. Included is the spectral theory for compact self adjoint operators with a wide range of applications. Part of the volume is devoted to Banach spaces and operators acting on these spaces. The theory is richly illustrated by many examples including Laurent, Toeplitz and Poincaré operators.
The present book is an expanded and enriched version ofthe textBasicOperator Theory, written by the first two authors more than twenty years ago. Since then the three ofus have used the basic operator theory text in various courses. This experience motivated us to update and improve the old text by including a wider variety ofbasic classes ofoperators and their applications. The present book has also been written in such a way that it can serve as an introduction to our previous booksClassesofLinearOperators, Volumes I and II. We view the three books as a unit. We gratefully acknowledge the support of the mathematical departments of Tel-Aviv University, the University of Maryland at College Park, and the Vrije Universiteit atAmsterdam. The generous support ofthe Silver Family Foundation is highly appreciated. Amsterdam, November 2002 The authors Introduction This elementary text is an introduction to functional analysis, with a strong emphasis on operator theory and its applications. It is designed for graduate and senior undergraduate students in mathematics, science, engineering, and other fields. Introduction.- Hilbert Spaces.- Bounded Linear Operators on Hilbert Spaces.- Laurent and Toeplitz Operators on Hilbert Spaces.- Spectral Theory of Compact Self-Adjoint Operators.- Spectral Theory of Integral Operators.- Unbounded Operators on Hilbert Spaces.- Oscillations of an Elastic String.- Operational Calculus with Applications.- Solving Linear Equations by Iterative Methods.- Further Developments of the Spectral Theorem.- Banach Spaces.- Linear Operators Banach Spaces.- Compact Operators on Banach Spaces.- Poincaré Operators: Determinant and Trace.- Fredholm Operators.- Toeplitz and Singular Integral Operators.- Nonlinear Operators.
"With a name like Gohberg-Goldberg-Kaashoek, it has got to be good. But let me count the ways. If you are interested in learning the basic theories of Hilbert and Banach spaces together with the well-known operators that act on them, this book is for you. It is intended for advanced undergraduate and beginning graduate students in mathematics.... The volume is actually a second edition of Basic Operator Theory by Gohberg and Goldberg.... Exercises and examples have been added.... The text is well written, clear, and readable.... A student at the specified level will have no difficulty understanding the presentation."
-SIAM Review
"The material is presented in a very clear way, emphasizing concepts over technical generalities. Another pleasant feature of the book is the large number of examples and exercises. In summary, the authors have successfully found an interesting way from the elementary level to some of the highlights of operator theory."
- Monatshefte für Mathematik
