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实分析：分析综合教程（第1部分）（影印版）

样章
作者：
Barry Simon
定价：
269.00元
ISBN：
978-7-04-059306-8
版面字数：
1350.000千字
开本：
特殊
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2023-03-24
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析
暂无
  • 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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