实数学分析（影印版）
2009-2
高等教育出版社
Charles Chapman Pugh
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本书是作者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。本书内容包括实数、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。
1 Real Numbers 1 Preliminaries 2 Cuts 3 Euclidean Space 4 Cardinality 5* Comparing Cardinalities 6* The Skeleton of Calculus Exercises2 A Taste of Topology 1 Metric Space Concepts 2 Compactness 3 Connectedness 4 Coverings 5 Cantor Sets 6* Cantor Set Lore 7* Completion Exercises3 Functions of a Real Variable 1 Differentiation 2 Riemann Integration 3 Series Exercises4 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 ODE's 6* Analytic Functions 7* Nowhere Differentiable Continuous Functions 8* Spaces of Unbounded Functions Exercises5 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 Cavalieri's Principle Appendix C: A Short Excursion into the Complex Field Appendix D: Polar Form Appendix E: Determinants Exercises6 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 Lebesgue's Fundamental Theorem of Calculus 9 Lebesgue's Last Theorem Appendix A: Translations and Nonmeasurable sets Appendix B: The Banach-Tarski Paradox Appendix C: Riemann integrals as undergraphs Appendix D: Littlewood's Three Principles Appendix E: Roundness Appendix F: Money Suggested Reading Bibliography ExercisesIndex
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評《實數學分析》在這部大學實分析的新引論中,作者採用了不同的寫法,即強調圖片在數學和難題中的重要性.其闡述是非正規的和輕鬆的,有很多有用的旁白,例子,和數學家的偶然間的評論.本書強調定理的理解而非正規的證明.本書包含500多個認真製作的練習,從簡單的練習到挑戰性的練習都有(因此本教材可用作實分析的例題和練習的資料書).
全英语的，看起来有点吃力，在努力中，感觉很可以的
此书内容上虽然跟国内实变函数教材有交集，但作者序言中说明此书的预期读者是“budding pure mathematician"(我译成”含苞待放的纯数学家“)，而且暗示先导课是普通的微积分，证明此书就是美国的”高等微积分“教材，应该译为”真正的数学分析“或者”数学分析当如此“lol..
数学分析当如此---
对这评论实在没法吐槽
楼主还是不要看这书了……