實數(shù)學(xué)分析（影印版）

內(nèi)容概要

本書是作者Pugh在伯克利大學(xué)講授數(shù)學(xué)分析課程30多年之久的基礎(chǔ)上編寫而成，書中語言表述生動活潑、通俗易懂，引用了很多有價值的例子以及來自Dieudonne，Littlewood和Osserman等幾位數(shù)學(xué)家的評論，還精心挑選了500多個精彩的練習(xí)題。本書內(nèi)容包括實數(shù)、拓?fù)渲R初步、實變函數(shù)、函數(shù)空間、多元微積分、Lebesgue積分理論等，其中多元微積分的講法較為接近當(dāng)前數(shù)學(xué)界常用的語言，將會對我國數(shù)學(xué)分析的教學(xué)產(chǎn)生積極的影響。

書籍目錄

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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