This book contains almost 450 exercises, all with complete solutions; it provides supplementary examples, counter-examples, and applications for the basic notions usually presented in an introductory course in Functional Analysis. Three comprehensive sections cover the broad topic of functional analysis. A large number of exercises on the weak topologies is included. Contents......Page 3 Preface......Page 5 Some Standard Notations and Conventions......Page 7 Part I: Normed spaces......Page 9 1. Open, closed, and bounded sets in normed spaces......Page 10 1.1 Exercises......Page 11 1.2 Solutions......Page 18 2. Linear and continuous operators on normed spaces......Page 43 2.1 Exercises......Page 44 2.2 Solutions......Page 49 3.1 Exercises......Page 75 3.2 Solutions......Page 79 4.1 Exercises......Page 93 4.2 Solutions......Page 99 5. Compactness in Banach spaces. Compact operators......Page 114 5.1 Exercises......Page 115 5.2 Solutions......Page 122 6.1 Exercises......Page 154 6.2 Solutions......Page 162 7.1 Exercises......Page 182 7.2 Solutions......Page 187 8. Applications for the Hahn-Banach theorem......Page 202 8.1 Exercises......Page 203 8.2 Solutions......Page 206 9. Baire's category. The open mapping and closed graph theorems......Page 220 9.1 Exercises......Page 221 9.2 Solutions......Page 227 Part II: Hilbert spaces......Page 248 10. Hilbert spaces, general theory......Page 249 10.1 Exercises......Page 251 10.2 Solutions......Page 256 11.1 Exercises......Page 277 11.2 Solutions......Page 287 12. Linear and continuous operators on Hilbert spaces......Page 311 12.1 Exercises......Page 312 12.2 Solutions......Page 324 Part III: General topological spaces......Page 372 13. Linear topological and locally convex spaces......Page 373 13.1 Exercises......Page 374 13.2 Solutions......Page 382 14. The weak topologies......Page 408 14.1 Exercises......Page 410 14.2 Solutions......Page 417 Bibliography......Page 449 List of Symbols......Page 452 Index......Page 454 The understanding of results and notions for a student in mathematics requires solving ex ercises. The exercises are also meant to test the reader's understanding of the text material, and to enhance the skill in doing calculations. This book is written with these three things in mind. It is a collection of more than 450 exercises in Functional Analysis, meant to help a student understand much better the basic facts which are usually presented in an introductory course in Functional Analysis. Another goal of this book is to help the reader to understand the richness of ideas and techniques which Functional Analysis offers, by providing various exercises, from different topics, from simple ones to, perhaps, more difficult ones. We also hope that some of the exercises herein can be of some help to the teacher of Functional Analysis as seminar tools, and to anyone who is interested in seeing some applications of Functional Analysis. To what extent we have managed to achieve these goals is for the reader to decide. This Book Of Exercises In Functional Analysis Contains Almost 450 Exercises (all With Complete Solutions), Providing Supplementary Examples, Counter-examples And Applications For The Basic Notions Usually Presented In An Introductory Course In Functional Analysis. It Contains Three Parts. The First One Contains Exercises On The General Properties For Sets In Normed Spaces, Linear Bounded Operators On Normed Spaces, Reflexivity, Compactness In Normed Spaces, And On The Basic Principles In Functional Analysis: The Hahn-banach Theorem, The Uniform Boundedness Principle, The Open Mapping And The Closed Graph Theorems. The Second One Contains Exercises On The General Theory Of Hilbert Spaces, The Riesz Representation Theorem, Orthogonality In Hilbert Spaces, The Projection Theorem And Linear Bounded Operators On Hilbert Spaces. The Third One Deals With Linear Topological Spaces, And Includes A Large Number Of Exercises On The Weak Topologies.
