Elements of the Theory of Functions and Functional Analysis, Volume 1, Metric and Normed Spaces

Cover......Page 1
S Title......Page 2
OTHER GRAYLOCK PUBLICATIONS......Page 3
Title: Elements of the Theory of Functionsand Functional Analysis, VOLUME 1, METRIC AND NORMED SPACES......Page 4
LCCN 570014021......Page 5
CONTENTS......Page 6
PREFACE......Page 8
TRANSLATOR'S NOTE......Page 10
§1. The concept of set. Operations on sets......Page 12
§2. Finite and infinite sets. Denumerability......Page 14
§3. Equivalence of sets......Page 17
§4. Nondenumerability of the set of real numbers......Page 19
§5. The concept of cardinal number......Page 20
§6. Partition into classes......Page 22
§7. Mappings of sets. General concept of function......Page 24
§8. Definition and examples of metric spaces......Page 27
§9. Convergence of sequences. Limit points......Page 34
§10. Open and closed sets......Page 37
§11. Open and closed sets on the real line......Page 42
§12. Continuous mappings. Homeomorphism. Isometry......Page 44
§13. Complete metric spaces......Page 47
§14. Principle of contraction mappings and its applications......Page 54
§ 15. Applications of the principle of contraction mappings in analysis......Page 57
§16. Compact sets in metric spaces......Page 62
§17. ArzeIa's theorem and its applications......Page 64
§18. Compacta......Page 68
§19. Real functions in metric spaces......Page 73
§20. Continuous curves in metric spaces......Page 77
§21. Definition and examples of normed linear spaces......Page 82
§22. Convex sets in normed linear spaces......Page 85
§23. Linear functionals......Page 88
§24. The conjugate space......Page 92
§25. Extension of linear functionals......Page 97
§26. The second conjugate space......Page 99
§27. Weak convergence......Page 101
§28. Weak convergence of linear functionals......Page 103
§29. Linear operators......Page 106
Generalized Functions......Page 116
§30. Spectrum of an operator. Resolvents......Page 121
§31. Completely continuous operators......Page 123
§32. Ljnear operator equations. The Fredholm theorems......Page 127
LIST OF SYMBOLS......Page 133
LIST OF THEOREMS......Page 134
BASIC LITERATURE......Page 136
INDEX......Page 138