Functional Analysis and Operator Theory
Description
The book contains a collection of more than 800 problems from all main chapters of functional analysis, with theoretical background and solutions. It is mostly intended for undergraduate students who are starting to study the course of functional analysis. The book will also be useful for graduate and post- graduate students and researchers who wish to refresh their knowledge and deepen their understanding of the subject, as well as for teachers of functional analysis and related disciplines. It can be used for independent study as well. It is assumed that the reader has mastered standard courses of calculus and measure theory and has basic knowledge of linear algebra, analytic geometry, and differential equations.
This collection of problems can help students of different levels of training and different areas of specialization to learn how to solve problems in functional analysis. Each chapter of the book has similar structure and consists of the following sections: Theoretical Background, Examples of Problems with Solutions, and Problems to Solve. The book contains theoretical preliminaries to ensure that the reader understands the statements of problems and is able to successfully solve them. Then examples of typical problems with detailed solutions are included, and this is relevant not only for those students who have significant difficulties in studying this subject, but also for other students who due to various circumstances ?could be deprived of communication with a teacher. There are problems for independent solving, and the corresponding selection of problems reflects all the main plot lines that relate to a given topic.
The number of problems is sufficient both for a teacher to give practical lessons, to set homework, to prepare tasks for various forms of control, and for those students who want to study the discipline more deeply. Problems of a computational nature are provided with answers, while theoretical problems, the solutions ofwhich require non-trivial ideas or new techniques, are provided with detailed hints or solutions to introduce the reader to the corresponding ideas or techniques.Reviews / Votes
"The book is carefully written, with illuminating introductory theoretical parts and well chosen problems with hints and solutions. It will be of help to all students studying functional analysis or related fields as well as an auxiliary instrument in the teaching process." (Stefan Cobzas, zbMATH 1557.46001, 2025)
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Persons
Dr. Volodymyr Brayman received his PhD in probability and statistics from the Institute of Mathematics of National Academy of Sciences of Ukraine in 2007. He is currently an Assistant Professor at the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv. He is an expert in stochastic processes, a jury member in various mathematical competitions, and an author of numerous problems proposed at the competitions. He co-authored with Prof. Alexander Kukush a problem book titled «Undergraduate Mathematics Competitions (1995-2016)», Taras Shevchenko National University of Kyiv (ISBN 978-3-319-58672-4), published with Springer.
Prof. Andrii Chaikovskyi received his PhD in differential equations from Taras Shevchenko National University of Kyiv in 2001 and completed his postdoctoral degree in differential equations (Habilitation) in 2012. He is currently the Head of the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv. He is the author/coauthor of more than 60 research papers. His research interests include abstract differential and difference equations and approximation theory.
Dr. Oleksii Konstantinov is an Associate Professor of the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv. His research interests include Operator Theory, Mathematical Scattering Theory and Differential Operators.
Prof. Alexander Kukush received his PhD in probability and statistics from Kyiv University in 1982 and completed his postdoctoral degree in probability and statistics (Habilitation) in 1995. He is a Professor of the Department of Mathematical Analysis at Taras Shevchenko National University of Kyiv. He is the author/coauthor of more than 150 research papers and 4 books. His research interests include mathematical and applied statistics, actuarial and financial mathematics.
Prof. Yuliya Mishura received her PhD in probability and statistics from Kyiv University in 1978 and completed her postdoctoral degree in probability and statistics (Habilitation) in 1990. She is currently a Professor of the Department of Probability, Statistics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. She is the author/coauthor of more than 300 research papers and 12 books. Her research interests include theory and statistics of stochastic processes, stochastic differential equations, fractional calculus and fractional processes, stochastic analysis. Dr. Oleksii Nesterenko received his PhD in mathematical analysis from Taras Shevchenko National University of Kyiv in 2007. He is currently an Associate Professor of the Department of Mathematical Analysis of Taras Shevchenko National University of Kyiv. He is the author/coauthor of more than 10 research papers. His research interests include approximation theory.
Content
Preface.- Banach Spaces.- Hilbert Spaces.- Continuous Linear Functionals.- Hahn - Banach Theorem.- Weak and Weak* Convergence.- Bounded Linear Operators.- Uniform, Strong and Weak Operator Convergences.- Inverse Operators.- Classes of Linear Operators in Hilbert Space.- Compact Sets and Operators.- Spectrum of Linear Operators.- Spectral Theory of Compact Operators.- Integral Equations.- Generalized Functions.- Answers, hints and solutions.- List of notations.- References.