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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 сcould 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.
Preface
List of Notations
Banach Spaces
Theoretical Background
Basic Examples of Banach Spaces
Examples of Problems with Solutions
Problems to Solve
Hilbert Spaces
Theoretical Background
Examples of Hilbert Spaces
Examples of Problems with Solutions
Problems to Solve
Continuous Linear Functionals
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Hahn–Banach Theorem
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Weak and Weak-* Convergence
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Bounded Linear Operators
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Uniform, Strong and Weak Operator Convergence
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Inverse Operators
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Classes of Linear Operators in Hilbert Space
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Compact Sets and Operators
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Spectrum of Linear Operators
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Spectral Theory of Compact Operators
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Integral Equations
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Generalized Functions
Theoretical Background
Examples of Problems with Solutions
Problems to Solve
Answers, Hints and Solutions
References