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

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## A.N. Kolmogorov and S.V. Fomin:

## 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

### 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**

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces