实分析原理(第3版)(Principles of real analysis)
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品牌: 阿里普兰蒂斯
基本信息·出版社：世界图书出版公司
·页码：415 页
·出版日期：2009年
·ISBN：7506292726/9787506292726
·条形码：9787506292726
·包装版本：3版
·装帧：平装
·开本：24
·正文语种：英语
·外文书名：Principles of real analysis
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内容简介This is the third edition of Principles of Real Alysis, first published in 1981. The aim of this edition is to accommodate the current needs for the traditional real analysis course that is usually taken by the senior undergraduate or by the first year graduate student in mathematics. This edition differs substantially from the second edition. Each chapter has been greatly improved by incorporating new material and by rearranging the old material. Moreover, a new chapter （Chapter 6） on Hilbert spaces and Fourier analysis has been added.
目录
Preface
CHAPTER 1. FUNDAMENTALS OF REAL ANALYSIS
1. Elementary Set Theory
2. Countable and Uncountable Sets
3. The Real Numbers
4. Sequences of Real Numbers
5. The Extended Real Numbers
6. Metric Spaces
7. Compactness in Metric Spaces
CHAPTER 2. TOPOLOGY AND CONTINUITY
8. Topological Spaces
9. Continuous Real-Valued Functions
10. Separation Properties of Continuous Functions
11. The Stone-Weierstrass Approximation Theorem
CHAPTER 3. THE THEORY OF MEASURE
12. Semirings and Algebras of Sets
13. Measures on Semirings
14. Outer Measures and Measurable Sets
15. The Outer Measure Generated by a Measure
16. Measurable Functions
17. Simple and Step Functions
18. The Lebesgue Measure
19. Convergence in Measure
20. Abstract Measurability
CHAPTER 4. THE LEBESGUE INTEGRAL
21. Upper Functions
22. Integrable Functions
23. The Riemann Integral as a Lebesgue Integral
24. Applications of the Lebesgue Integral
25. Approximating Integrable Functions
26. Product Measures and Iterated Integrals
CHAPTER 5. NORMED SPACES AND Lp-SPACES
27. Normed Spaces and Banach Spaces
28. Operators Between Banach Spaces
29. Linear Functionals
30. Banach Lattices
31. Lp-Spaces
CHAPTER 6. HILBERT SPACES
32. Inner Product Spaces
33. Hilbert Spaces
34. Orthonormal Bases
35. Fourier Analysis
CHAPTER 7. SPECIAL TOPICS IN INTEGRATION
36. Signed Measures
37. Comparing Measures and the
Radon-Nikodym Theorem
38. The Riesz Representation Theorem
39. Differentiation and Integration
40. The Change of Variables Formula
Bibliography
List of Symbols
Index
……[看更多目录]
序言This is the third edition of Principles of Real Alysis, first published in 1981. The aim of this edition is to accommodate the current needs for the traditional real analysis course that is usually taken by the senior undergraduate or by the first year graduate student in mathematics. This edition differs substantially from the second edition. Each chapter has been greatly improved by incorporating new material and by rearranging the old material. Moreover, a new chapter （Chapter 6） on Hilbert spaces and Fourier analysis has been added.
The subject matter of the book focuses on measure theory and the Lebesgue integral as well as their applications to several functional analytic directions. As in the previous editions, the presentation of measure theory is built upon the notion of a semiring in connection with the classical Carath6odory extension procedure. We believe that this natural approach can be easily understood by the student. An extra bonus of the presentation of measure theory via the semirmg approach is the fact that the product of semirings is always a semiring while the product of 0r-algebras is a semiring but not a o-algebra. This simple but important fact demonstrates that the semiring approach is the natural setting for product measures and iterated integrals.
The theory of integration is also studied in connection with partially ordered vector spaces and, in particular, in connection with the theory of vector lattices. The theory of vector lattices provides the natural framework for formalizing and interpreting the basic properties of measures and integrals （such as the Radon- Nikodym theorem, the Le be sgue and Jordan decompositions of a measure, and the Riesz representation theorem）. The bibliography at the end of the book includes several books that the reader can consult for further reading and for different approaches to the presentation of measure theory and integration.
In order to supplement the learning effort, we have added many problems （more than 150 for a total of 609） of varying degrees of difficulty. Students who solve a good percentage of these problems will certainly master the material of this book. To indicate to the reader that the development of real analysis was a collective effort by many great scientists from several countries and continents through the ages, we have included brief biographies of all contributors to the subject mentioned in this book.
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