# Book:A.N. Kolmogorov/Elements of the Theory of Functions and Functional Analysis/Volume 1

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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, Volume 1*

Published $1957$, **Graylock Press** (translated by Leo F. Boron).

**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**