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)

Subject Matter

Functional Analysis

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