Functional AnalysisFunctional Analysis is primarily concerned with the structure of infinite dimensional vector spaces and the transformations, which are frequently called operators, between such spaces. The elements of these vector spaces are usually functions with certain properties, which map one set into another. Functional analysis became one of the success stories of mathematics in the 20th century, in the search for generality and unification. Although it remains a very attractive field of pure mathematics, it has also proven to be an indispensable and powerful tool for physicists, engineers and economists involved in research and development, for helping them understand their subject in depth. This book is designed to provide the reader with a solid foundation of almost the entire spectrum of functional analysis, upon which each reader may build their own special structure, tailored to his or her purposes. The only prerequisite is the familiarity with the classical analysis of standard level. The book then provides a smooth but fast-paced systematical passage to the required advanced mathematical level, without sacrificing the mathematical rigor with almost no reference to outside materials to offering also a complete coverage of this discipline. We believe that the latter is one of the unique features of this book. None of the books in literature cover that much material, starting from a rather modest mathematical level. Functional Analysis will be primarily of interest to graduate students in applied mathematics, in all branches of engineering, in physical and economical sciences and to those working in research and development in industry and research institutes. |
Contents
11 SCOPE OF THE CHAPTER | 1 |
12 SETS | 2 |
13 SET OPERATIONS | 6 |
14 CARTESIAN PRODUCT RELATIONS | 12 |
15 FUNCTIONS | 14 |
16 INVERSE FUNCTIONS | 21 |
17 PARTIAL ORDERING | 26 |
18 EQUIVALENCE RELATION | 31 |
57 COMPACT METRIC SPACES | 319 |
58 APPROXIMATION | 330 |
59 THE SPACE OF FRACTALS | 340 |
V EXERCISES | 350 |
61 SCOPE OF THE CHAPTER | 357 |
62 NORMED SPACES | 358 |
63 SEMINORMS | 366 |
64 SERIES OF VECTORS | 374 |
19 OPERATIONS ON SETS | 35 |
110 CARDINALITY OF SETS | 38 |
111 ABSTRACT MATHEMATICAL SYSTEMS | 50 |
112 VARIOUS ABSTRACT SYSTEMS | 54 |
I EXERCISES | 66 |
21 SCOPE OF THE CHAPTER | 71 |
22 LINEAR VECTOR SPACES | 72 |
23 SUBSPACES | 74 |
24 LINEAR INDEPENDENCE AND DEPENDENCE | 80 |
25 BASIS AND DIMENSION | 83 |
26 TENSOR PRODUCT OF LINEAR SPACES | 89 |
27 LINEAR TRANSFORMATIONS | 91 |
28 MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS | 99 |
29 EQUIVALENT AND SIMILAR LINEAR TRANSFORMATIONS | 102 |
210 LINEAR FUNCTIONALS ALGEBRAIC DUAL | 106 |
211 LINEAR EQUATIONS | 120 |
212 EIGENVALUES AND EIGENVECTORS | 121 |
II EXERCISES | 147 |
31 SCOPE OF THE CHAPTER | 157 |
32 PROPERTIES OF SETS OF REAL NUMBERS | 158 |
33 COMPACTNESS | 167 |
34 SEQUENCES | 171 |
35 LIMIT AND CONTINUITY IN FUNCTIONS | 178 |
36 DIFFERENTIATION AND INTEGRATION | 182 |
37 MEASURE OF A SET LEBESGUE INTEGRAL | 189 |
III EXERCISES | 217 |
41 SCOPE OF THE CHAPTER | 221 |
42 TOPOLOGICAL STRUCTURE | 222 |
43 BASES AND SUBBASES | 232 |
44 SOME TOPOLOGICAL CONCEPTS | 234 |
45 NUMERICAL FUNCTIONS | 252 |
46 TOPOLOGICAL VECTOR SPACES | 256 |
IV EXERCISES | 257 |
51 SCOPE OF THE CHAPTER | 261 |
52 THE METRIC AND THE METRIC TOPOLOGY | 262 |
53 VARIOUS METRIC SPACES | 267 |
54 TOPOLOGICAL PROPERTIES OF METRIC SPACES | 281 |
55 COMPLETENESS OF METRIC SPACES | 292 |
56 CONTRACTION MAPPINGS | 312 |
65 BOUNDED LINEAR OPERATORS | 376 |
66 EQUIVALENT NORMED SPACES | 398 |
67 BOUNDED BELOW OPERATORS | 404 |
68 CONTINUOUS LINEAR FUNCTIONALS | 405 |
69 TOPOLOGICAL DUAL | 414 |
610 STRONG AND WEAK TOPOLOGIES | 450 |
611 COMPACT OPERATORS | 456 |
612 CLOSED OPERATORS | 467 |
613 CONJUGATE OPERATORS | 473 |
614 CLASSIFICATION OF CONTINUOUS OPERATORS | 487 |
VI EXERCISES | 489 |
71 SCOPE OF THE CHAPTER | 499 |
72 INNER PRODUCT SPACES | 500 |
73 ORTHOGONAL SUBSPACES | 511 |
74 ORTHONORMAL SETS AND FOURIER SERIES | 516 |
75 DUALS OF HILBERT SPACES | 531 |
76 LINEAR OPERATORS IN HILBERT SPACES | 536 |
77 FORMS AND VARIATIONAL EQUATIONS | 557 |
VII EXERCISES | 564 |
81 SCOPE OF THE CHAPTER | 573 |
82 THE RESOLVENT SET AND THE SPECTRUM | 574 |
83 THE RESOLVENT OPERATOR | 575 |
84 THE SPECTRUM OF A BOUNDED OPERATOR | 579 |
85 THE SPECTRUM OF A COMPACT OPERATOR | 581 |
86 FUNCTIONS OF OPERATORS | 584 |
87 SPECTRAL THEORY IN HILBERT SPACES | 593 |
VIII EXERCISES | 608 |
91 SCOPE OF THE CHAPTER | 613 |
93 HIGHER ORDER FRECHET DERIVATIVES | 627 |
94 INTEGRATION OF OPERATORS | 629 |
95 THE METHOD OF NEWTON | 638 |
96 THE METHOD OF STEEPEST DESCENT | 650 |
97 THE IMPLICIT FUNCTION THEOREM | 670 |
IX EXERCISES | 674 |
| 677 | |
| 680 | |
| 681 | |
| 683 | |