实数学分析

1 Real Numbers

1 Preliminaries

2 Cuts

3 Euclidean Space

4 Cardinality

5* Comparing Cardinalities

6* The Skeleton of Calculus

Exercises

2 A Taste of Topology

1 Metric Space Concepts

2 Compactness

3 Connectedness

4 Coverings

5 Cantor Sets

6* Cantor Set Lore

7* Completion

Exercises

3 Functions of a Real Variable

1 Differentiation

2 Riemann Integration

3 Series

Exercises

4 Function Spaces

1 Uniform Convergence and C0[a, b]

2 Power Series

3 Compactness and Equicontinuity in CO

4 Uniform Approximation in Co

5 Contractions and ODEs

6* Analytic Functions

7* Nowhere Differentiable Continuous Functions

8* Spaces of Unbounded Functions

Exercises

5 Multivariable Calculus

1 Linear Algebra

2 Derivatives

3 Higher derivatives

4 Smoothness Classes

5 Implicit and Inverse Functions

6* The Rank Theorem

7* Lagrange Multipliers

8 Multiple Integrals

9 Differential Forms

10 The General Stokes Formula

11* The Brouwer Fixed Point Theorem

Appendix A: Perorations of Dieudonne

Appendix B: The History of Cavalieris Principle

Appendix C: A Short Excursion into

the Complex Field

Appendix D: Polar Form

Appendix E: Determinants

Exercises

6 Lebesgue Theory

1 Outer measure

2 Measurability

3 Regularity

4 Lebesgue integrals

5 Lebesgue integrals as limits

6 Italian Measure Theory

7 Vitali coverings and density points

8 Lebesgues Fundamental Theorem of Calculus

9 Lebesgues Last Theorem

Appendix A: Translations and Nonmeasurable sets

Appendix B: The Banach-Tarski Paradox

Appendix C: Riemann integrals as undergraphs

Appendix D: Littlewoods Three Principles

Appendix E: Roundness

Appendix F: Money

Suggested Reading

Bibliography

Exercises

Index

作者Pugh在伯克利大学讲授数学分析课程30多年之久的基础上编写而成，书中语言表述生动活泼、通俗易懂，引用了很多有价值的例子以及来自Dieudonne，Littlewood和Osserman等几位数学家的评论，还精心挑选了500多个精彩的练习题。《实数学分析（影印版）》内容包括实数、拓扑知识初步、实变函数、函数空间、多元微积分、Lebesgue积分理论等，其中多元微积分的讲法较为接近当前数学界常用的语言，将会对我国数学分析的教学产生积极的影响。