实分析原理
2009-1
世界图书出版公司
(美) (阿里普兰蒂斯Aliprantis) (C.D)
415
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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.
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.
PrefaceCHAPTER 1. FUNDAMENTALS OF REAL ANALYSIS1. Elementary Set Theory2. Countable and Uncountable Sets3. The Real Numbers4. Sequences of Real Numbers5. The Extended Real Numbers6. Metric Spaces7. Compactness in Metric SpacesCHAPTER 2. TOPOLOGY AND CONTINUITY8. Topological Spaces9. Continuous Real-Valued Functions10. Separation Properties of Continuous Functions11. The Stone-Weierstrass Approximation TheoremCHAPTER 3. THE THEORY OF MEASURE12. Semirings and Algebras of Sets13. Measures on Semirings14. Outer Measures and Measurable Sets15. The Outer Measure Generated by a Measure16. Measurable Functions17. Simple and Step Functions18. The Lebesgue Measure19. Convergence in Measure20. Abstract MeasurabilityCHAPTER 4. THE LEBESGUE INTEGRAL21. Upper Functions22. Integrable Functions23. The Riemann Integral as a Lebesgue Integral24. Applications of the Lebesgue Integral25. Approximating Integrable Functions26. Product Measures and Iterated IntegralsCHAPTER 5. NORMED SPACES AND Lp-SPACES27. Normed Spaces and Banach Spaces28. Operators Between Banach Spaces29. Linear Functionals30. Banach Lattices31. Lp-SpacesCHAPTER 6. HILBERT SPACES32. Inner Product Spaces33. Hilbert Spaces34. Orthonormal Bases35. Fourier AnalysisCHAPTER 7. SPECIAL TOPICS IN INTEGRATION36. Signed Measures37. Comparing Measures and theRadon-Nikodym Theorem38. The Riesz Representation Theorem39. Differentiation and Integration40. The Change of Variables FormulaBibliographyList of SymbolsIndex
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