Exercises in Functional Analysis

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