Elements of the Theory of Functions and Functional Analysis

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Elements of the Theory of Functions and Functional Analysis

Kolmogorov, A. N.

ウェブストア価格 ¥5,181（本体¥4,710）
Dover Publications Inc.（2003/03発売）
外貨定価 US$ 25.00
ポイント 47pt

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製本 Paperback:紙装版/ペーパーバック版／ページ数 288 p.
言語 ENG
商品コード 9780486406831
DDC分類 510

基本説明

Originally published in two volumes in 1957. This advanced-level text is based on courses and lectures given by the authors at Moscow State University and the University of Moscow.

Full Description

Originally published in two volumes, this advanced-level text is based on courses and lectures given by the authors at Moscow State University and the University of Moscow.

Reprinted here in one volume, the first part is devoted to metric and normal spaces. Beginning with a brief introduction to set theory and mappings, the authors offer a clear presentation of the theory of metric and complete metric spaces. The principle of contraction mappings and its applications to the proof of existence theorems in the theory of differential and integral equations receives detailed analysis, as do continuous curves in metric spaces - a topic seldom discussed in textbooks.

Part One also includes discussions of other subjects, such as elements of the theory of normed linear spaces, weak sequential convergence of elements and linear functionals, adjoint operators, and linear operator equations.

Part Two focuses on an exposition of measure theory, the Lebesque interval and Hilbert Space. Both parts feature numerous exercises at the end of each section and include helpful lists of symbols, definitions, and theorems.

One-volume reprint of the two-volume edition published by the Graylock Press, Rochester, New York, 1957.

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