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: 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