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

A.N. Kolmogorov and S.V. Fomin: Elements of the Theory of Functions and Functional Analysis, Volume $\text { 1 }$

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

Subject Matter

• Functional Analysis

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

Cited by

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.)
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces