# Book:A.N. Kolmogorov/Elements of the Theory of Functions and Functional Analysis

## A.N. Kolmogorov and S.V. Fomin: Elements of the Theory of Functions and Functional Analysis

Published $\text {1957 - 1961}$, Graylock Press (translated by Leo F. Boron)

### Volume 1 Contents

Preface
Translator's Note
Chapter I: Fundamentals of Set Theory
1. The Concept of Set. Operations on Sets
2. Finite and Infinite Sets. Denumerability
3. Equivalence of Sets
4. The Nondenumerability of the Set of Real Numbers
5. The Concept of Cardinal Number
6. Partition into Classes
7. Mappings of Sets. General Concept of Function
Chapter II: Metric Spaces
8. Definition and Examples of Metric Spaces
9. Convergence of Sequences. Limit Points
10. Open and Closed Sets
11. Open and Closed Sets on the Real Line
12. Continuous Mappings. Homeomorphism. Isometry
13. Complete Metric Spaces
14. The Principle of Contraction Mappings and its Applications
15. Applications of the Principle of Contraction Mappings in Analysis
16. Compact Sets in Metric Spaces
17. Arzela's Theorem and its Applications
18. Compacta
19. Real Functions in Metric Spaces
20. Continuous Curves in Metric Spaces
Chapter III: Normed Linear Spaces
21. Definition and Examples of Normed Linear Spaces
22. Convex Sets in Normed Linear Spaces
23. Linear Functionals
24. The Conjugate Space
25. Extension of Linear Functionals
26. The Second Conjugate Space
27. Weak Convergence
28. Weak Convergence of Linear Functionals
29. Linear Operators
Addendum to Chapter III: Generalized Functions
Chapter IV: Linear Operator Equations
30. Spectrum of an Operator. Resolvents
31. Completely Continuous Operators
32. Linear Operator Equations. Fredholm's Theorems
List of Symbols
List of Definitions
List of Theorems
Basic Literature
Index

### Volume 2 Contents

Preface
Translator's Note
Chapter V: Measure Theory
33. The measure of plane sets
34. Collections of sets
35. Measures of semi-rings. Extension of a measure on a semi-ring to the minimal ring over the semi-ring
36. Extension of Jordan curve
37. Complete additivity. The general problem of the extension of measures
38. The Lebesgue extension of a measure defined on a semi-ring with unity
39. Extension of Lebesgue measures in the general case
Chapter VI: Measurable Functions
40. Definition and fundamental properties of measurable functions
41. Sequences of measurable functions. Various types of convergence
Chapter VII: The Lebesgue integral
42. The Lebesgue integral of simple functions
43. The general definition and fundamental properties of the Lebesgue integral
44. Passage to the limit under the Lebesgue integral
45. Comparison of the Lebesgue and Riemann integrals
46. Products of sets and measures
47. The representation of plane measure in terms of the linear measure of sections and the geometric definition of the Lebesgue integral
48. Fubini's theorem
49. The integral as a set function
Chapter VIII: Square Integrable Functions
50. The space $L_2$
51. Mean convergence. Dense subsets $L_2$
52. $L_2$ spaces with countable bases
53. Orthogonal sets of functions. Orthogonalization
54. Fourier series over orthogonal sets. The Riesz-Fischer theorem
55. Isomorphism of the spaces $L_2$ and $l_2$
Chapter IX: Abstract Hilbert Space. Integral Equations with Symmetric Kernel
56. Abstract Hilbert space
57. Subspaces. Orthogonal complements. Direct sums
58. Linear and bilinear functionals in Hilbert space
59. Completely continuous self adjoint-operators in $H$
60. Linear operator equations with completely continuous operators
61. Integral equations with symmetric kernel
Supplement and Corrections to Volume 1
Index