A. N. Kolmogorov (1903-87) was a prominent Russian mathematician who made significant contributions to many areas of math, including probability theory, topology, logic, turbulence, and mechanics. In his many years on the faculty of Moscow State University, Dr. Kolmogorv's doctoral students included several who became prominent 20th-century mathematicians.

• 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

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A. N. Kolmogorov (1903-87) was a prominent Russian mathematician who made significant contributions to many areas of math, including probability theory, topology, logic, turbulence, and mechanics. In his many years on the faculty of Moscow State University, Dr. Kolmogorv's doctoral students included several who became prominent 20th-century mathematicians.

• 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

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A. N. Kolmogorov (1903-87) was a prominent Russian mathematician who made significant contributions to many areas of math, including probability theory, topology, logic, turbulence, and mechanics. In his many years on the faculty of Moscow State University, Dr. Kolmogorv's doctoral students included several who became prominent 20th-century mathematicians.

• 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

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