內容簡介
《實分析原理(第3版)》主要內容：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.
本書主要用統一、聯繫的觀點看待現代分析，把現代分析的不同分支領域視為高度相互聯繫而非分離的學科。通過這些聯繫可以使讀者在整體上對現代分析這一學科有更好的理解。
目錄
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
