Exercises in Functional AnalysisThe 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. |
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Contents
Open closed and bounded sets in normed spaces | 3 |
11 Exercises | 4 |
12 Solutions | 11 |
Linear and continuous operators on normed spaces | 36 |
21 Exercises | 37 |
22 Solutions | 42 |
Linear and continuous functional Reflexive spaces | 68 |
32 Solutions | 72 |
Baires category The open mapping and closed graph theorems | 213 |
92 Solutions | 220 |
Part II Hilbert spaces | 241 |
Hilbert spaces general theory | 243 |
101 Exercises | 245 |
102 Solutions | 250 |
The projection in Hilbert spaces | 271 |
Linear and continuous operators on Hilbert spaces | 305 |
The distance between sets in Banach spaces | 86 |
42 Solutions | 92 |
Compactness in Banach spaces Compact operators | 107 |
51 Exercises | 108 |
52 Solutions | 115 |
The Uniform Boundedness Principle | 147 |
62 Solutions | 155 |
The HahnBanach theorem | 175 |
72 Solutions | 180 |
Applications for the HahnBanach theorem | 195 |
82 Solutions | 199 |
121 Exercises | 306 |
122 Solutions | 318 |
Part III General topological spaces | 366 |
Linear topological and locally convex spaces | 368 |
132 Solutions | 377 |
The weak topologies | 403 |
142 Solutions | 412 |
| 444 | |
List of Symbols | 447 |
| 449 | |
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Common terms and phrases
addition an)nen apply Baire category balanced Banach space basis bounded Calculate called Cauchy chapter closed linear subspace compact complete consider continuous functional continuous operator contradiction convergent deduce defined definition denote dense element equivalent exercise fact finite finite-dimensional follows given Hahn-Banach hence Hilbert space hypothesis inequality Let H Let X limit linear and continuous linear space measure metric n E N neighborhood non-empty normed space observe obtain Obviously operator orthogonal projection positive Prove relatively compact respect scalar separable sequence solution subsequence subset theorem topology true uniformly unique weak whence
Bibliographic information
| Title | Exercises in Functional Analysis Volume 26 of Texts in the Mathematical Sciences, ISSN 0927-4529 |
| Authors | Constantin Costara, Dumitru Popa, D. Popa |
| Edition | illustrated |
| Publisher | Springer Science & Business Media, 2003 |
| ISBN | 1402015607, 9781402015601 |
| Length | 451 pages |
| Subjects | › Mathematics / Calculus Mathematics / Complex Analysis Mathematics / Functional Analysis Mathematics / Mathematical Analysis |
| Export Citation | BiBTeX EndNote RefMan |