实分析原理

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

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.