2012 Reprint of Volumes One and Two, 1957-1961. Exact facsimile of the original edition, not reproduced with Optical Recognition Software. A. N. Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, logic, turbulence, classical mechanics and computational complexity. Later in life Kolmogorov changed his research interests to the area of turbulence, where his publications beginning in 1941 had a significant influence on the field. In classical mechanics, he is best known for the Kolmogorov-Arnold-Moser theorem. In 1957 he solved a particular interpretation of Hilbert's thirteenth problem (a joint work with his student V. I. Arnold). He was a founder of algorithmic complexity theory, often referred to as Kolmogorov complexity theory, which he began to develop around this time. Based on the authors' courses and lectures, this two-part advanced-level text is now available in a single volume. Topics include metric and normed spaces, continuous curves in metric spaces, measure theory, Lebesque intervals, Hilbert space, and more. Each section contains exercises. Lists of symbols, definitions, and theorems.
Author(s): A. N. Kolmogorov, S. V. Fomin
Publisher: Martino Fine Books
Year: 2012
Language: English
Pages: 280
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Volume 1 - Metric and Normed Spaces ... 1
CONTENTS ... 6
PREFACE ... 8
TRANSLATOR'S NOTE ... 10
1 FUNDAMENTAL CONCEPTS OF SET THEORY ... 12
§1. The concept of set. Operations on sets ... 12
§2. Finite and infinite sets. Denumerability ... 14
§3. Equivalence of sets ... 17
§4. Nondenumerability of the set of real numbers ... 19
§5. The concept of cardinal number ... 20
§6. Partition into classes ... 22
§7. Mappings of sets. General concept of function ... 24
2 METRIC SPACES ... 27
§8. Definition and examples of metric spaces ... 27
§9. Convergence of sequences. Limit points ... 34
§10. Open and closed sets ... 37
§11. Open and closed sets on the real line ... 42
§12. Continuous mappings. Homeomorphism. Isometry ... 44
§13. Complete metric spaces ... 47
§14. Principle of contraction mappings and its applications ... 54
§ 15. Applications of the principle of contraction mappings in analysis ... 57
§16. Compact sets in metric spaces ... 62
§17. ArzeIa's theorem and its applications ... 64
§18. Compacta ... 68
§19. Real functions in metric spaces ... 73
§20. Continuous curves in metric spaces ... 77
3 NORMED LINEAR SPACES ... 82
§21. Definition and examples of normed linear spaces ... 82
§22. Convex sets in normed linear spaces ... 85
§23. Linear functionals ... 88
§24. The conjugate space ... 92
§25. Extension of linear functionals ... 97
§26. The second conjugate space ... 99
§27. Weak convergence ... 101
§28. Weak convergence of linear functionals ... 103
§29. Linear operators ... 106
ADDENDUM TO CHAPTER ITI ... 116
Generalized Functions ... 116
4 LINEAR OPERATOR EQUATIONS ... 121
§30. Spectrum of an operator. Resolvents ... 121
§31. Completely continuous operators ... 123
§32. Ljnear operator equations. The Fredholm theorems ... 127
LIST OF SYMBOLS ... 133
LIST OF DEFINITIONS ... 134
LIST OF THEOREMS ... 134
BASIC LITERATURE ... 136
INDEX ... 138
Volume 2 - Measure. The Lebesgue Integral. Hilbert Space ... 142Black,notBold,notItalic,open,FitWidth,-7
Cover ... 142
S Title ... 143
OTHER GRAYLOCK PUBLICATIONS ... 144
Title: Elements of the Theory of Functionsand Functional Analysis, VOLUME 2, MEASURE. THE LEBESGLTE INTEGRAL. HILBERT SPACE ... 145
Copyright ... 146
© 1961 GRAYLOCK PRESS ... 146
LCCN 5704134 ... 146
CONTENTS ... 147
PREFACE ... 149
TRANSLATORS' NOTE ... 151
Chapter V: MEASURE THEORY ... 152
§33. The measure of plane sets ... 152
§34. Collections of sets ... 166
EXERCISES ... 171
§35. Measures on semi-rings. Extension of a measure on a semi-ring to the minimal ring over the semi-ring ... 171
EXERCISES ... 173
§36. Extension of the Jordan measure ... 174
EXERCISES ... 178
§37. Complete additivity. The general problem of the extension of measures ... 179
EXERCISES ... 181
§38. The Lebesgue extension of a measure defined on a semi-ring with unity ... 182
EXERCISES ... 186
§39. Extension of Lebesgue measures in the general case ... 187
EXERCISES ... 188
Chapter VI: MEASURABLE FUNCTIONS ... 189
§40. Definition and fundamental properties of measurable functions ... 189
EXERCISES ... 193
§41. Sequences of measurable functions. Various types of convergence ... 193
EXERCISES ... 198
Chapter VII: THE LEBESGUE INTEGRAL ... 199
§42. The Lebesgue integral of simple functions ... 199
EXERCISES ... 201
§43. The general definition and fundamental properties of the Lebesgue integral ... 202
EXERCISES ... 206
§44. Passage to the limit under the Lebesgue integral ... 207
EXERCISES ... 212
§45. Comparison of the Lebesgue and Riemann integrals ... 213
EXERCISES ... 215
§46. Products of sets and measures ... 216
EXERCISES ... 219
§47. The representation of plane measure in terms of the linear measure of sections, and the geometric definition of the Lebesgue integral ... 219
EXERCISES ... 222
§48. Fubini's theorem ... 223
EXERCISES ... 226
§49. The integral as a set function ... 228
EXERCISES ... 229
Chapter VIII: SQUARE INTEGRABLE FUNCTIONS ... 230
§50. The space L2 ... 230
EXERCISES ... 233
§51. Mean convergence. Dense subsets of L2 ... 235
EXERCISES ... 238
§52. L2 spaces with countable bases ... 239
EXERCISES ... 241
§53. Orthogonal sets of functions. Orthogonalization ... 242
EXERCISES ... 246
§54. Fourier series over orthogonal sets. The Riesz-Fisher theorem ... 247
EXERCISES ... 251
§55. Isomorphism of the spaces L2 and 12 ... 252
EXERCISES ... 253
Chapter IX: SPACE. INTEGRAL EQUATIONS WITH SYMMETRIC KERNEL ... 254
§56. Abstract Hubert space ... 254
EXERCISES ... 256
§57. Subspaces. Orthogonal complements. Direct sums ... 257
EXERCISES ... 260
§58. Linear and bilinear functionals in Hubert space ... 261
EXERCISES ... 264
§59. Completely continuous seif-adjoint operators in H ... 266
EXERCISES ... 269
§60. Linear equations in completely continuous operators ... 270
§61. Integral equations with symmetric kernel ... 271
EXERCISES ... 273
SUPPLEMENT AND CORRECTIONS TO VOLUME 1 ... 274
INDEX ... 278
