实分析：分析综合教程（第1部分）（影印版）

Poincaré 奖得主 Barry Simon 的《分析综合教程》是一套五卷本的经典教程，可以作为研究生阶段的分析学教科书。这套分析教程提供了很多额外的信息，包含数百道习题和大量注释，这些注释扩展了正文内容并提供了相关知识的重要历史背景。阐述的深度和广度使这套教程成为几乎所有经典分析领域的宝贵参考资料。 第1部分致力于实分析。从一个角度来看，它将20世纪的微积分与极限积分（测度理论）和极限微分（分布理论）结合起来。另一方面，它展示了抽象空间的胜利：拓扑空间、Banach和Hilbert空间、测度空间、Riesz空间、Polish空间、局部凸空间、Fréchet空间、Schwartz空间和 L^(p )空间。最后是对大技巧的研究，包括Fourier级数和变换、对偶空间、Baire范畴、不动点定理、概率思想和Hausdorff维数。应用包括无处可微函数的构造、Brown 运动、空间填充曲线、矩问题的解、Harr测度和势理论中的平衡测度。 本书可供专业研究人员（数学家、部分应用数学家和物理学家）、讲授研究生阶段分析课程的教师以及在工作和学习中需要任何分析学知识的研究生阅读参考。

前辅文 Chapter 1. Preliminaries 1.1. Notation and Terminology 1.2. Metric Spaces 1.3. The Real Numbers 1.4. Orders 1.5. The Axiom of Choice and Zorn’s Lemma 1.6. Countability 1.7. Some Linear Algebra 1.8. Some Calculus Chapter 2. Topological Spaces 2.1. Lots of Definitions 2.2. Countability and Separation Properties 2.3. Compact Spaces 2.4. The Weierstrass Approximation Theorem and Bernstein Polynomials 2.5. The Stone–Weierstrass Theorem 2.6. Nets 2.7. Product Topologies and Tychonoff’s Theorem 2.8. Quotient Topologies Chapter 3. A First Look at Hilbert Spaces and Fourier Series 3.1. Basic Inequalities 3.2. Convex Sets, Minima, and Orthogonal Complements 3.3. Dual Spaces and the Riesz Representation Theorem 3.4. Orthonormal Bases, Abstract Fourier Expansions, and Gram–Schmidt 3.5. Classical Fourier Series 3.6. The Weak Topology 3.7. A First Look at Operators 3.8. Direct Sums and Tensor Products of Hilbert Spaces Chapter 4. Measure Theory 4.1. Riemann–Stieltjes Integrals 4.2. The Cantor Set, Function, and Measure 4.3. Bad Sets and Good Sets 4.4. Positive Functionals and Measures via L1(X) 4.5. The Riesz–Markov Theorem 4.6. Convergence Theorems 4.7. Comparison of Measures 4.8. Duality for Banach Lattices 4.9. Duality for Lp 4.10. Measures on Locally Compact and σ-Compact Spaces 4.11. Product Measures and Fubini’s Theorem 4.12. Infinite Product Measures and Gaussian Processes 4.13. General Measure Theory 4.14. Measures on Polish Spaces 4.15. Another Look at Functions of Bounded Variation 4.16. Bonus Section: Brownian Motion 4.17. Bonus Section: The Hausdorff Moment Problem 4.18. Bonus Section: Integration of Banach Space-Valued Functions 4.19. Bonus Section: Haar Measure on σ-Compact Groups Chapter 5. Convexity and Banach Spaces 5.1. Some Preliminaries 5.2. H¨older’s and Minkowski’s Inequalities: A Lightning Look 5.3. Convex Functions and Inequalities 5.4. The Baire Category Theorem and Applications 5.5. The Hahn–Banach Theorem 5.6. Bonus Section: The Hamburger Moment Problem 5.7. Weak Topologies and Locally Convex Spaces 5.8. The Banach–Alaoglu Theorem 5.9. Bonus Section: Minimizers in Potential Theory 5.10. Separating Hyperplane Theorems 5.11. The Krein–Milman Theorem 5.12. Bonus Section: Fixed Point Theorems and Applications Chapter 6. Tempered Distributions and the Fourier Transform 6.1. Countably Normed and Fr´echet Spaces 6.2. Schwartz Space and Tempered Distributions 6.3. Periodic Distributions 6.4. Hermite Expansions 6.5. The Fourier Transform and Its Basic Properties 6.6. More Properties of Fourier Transform 6.7. Bonus Section: Riesz Products 6.8. Fourier Transforms of Powers and Uniqueness of Minimizers in Potential Theory 6.9. Constant Coefficient Partial Differential Equations Chapter 7. Bonus Chapter: Probability Basics 7.1. The Language of Probability 7.2. Borel–Cantelli Lemmas and the Laws of Large Numbers and of the Iterated Logarithm 7.3. Characteristic Functions and the Central Limit Theorem 7.4. Poisson Limits and Processes 7.5. Markov Chains Chapter 8. Bonus Chapter: Hausdorff Measure and Dimension 8.1. The Carath´eodory Construction 8.2. Hausdorff Measure and Dimension Chapter 9. Bonus Chapter: Inductive Limits and Ordinary Distributions 9.1. Strict Inductive Limits 9.2. Ordinary Distributions and Other Examples of Strict Inductive Limits Bibliography Symbol Index Subject Index Author Index Index of Capsule Biographies

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