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

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