导语

  

内容提要

  

    本书是Folland教授的名著《实分析》的第二版。与第一版相比，在一些内容的编排上作了适当调整，同时引入了一些新的内容，去掉了已经过时的内容，更有利于学生学习与思考。作为一部优秀的教材，内容不仅涵盖了分析学的基本内容和技巧，还介绍了一些从事其他领域的研究工作所必需的基础知识。此外，教材中的大量习题，能够进一步拓展思维，从而易于更加深入地了解这些内容背后的真实想法。本书适用于理工类专业及相关专业的研究生。

Preface
0  Prologue
  0.1  The Language of Set Theory
  0.2  Orderings
  0.3  Cardinality
  0.4  More about Well Ordered Sets
  0.5  The Extended Real Number System
  0.6  Metric Spaces
  0.7  Notes and References
1  Measures
  1.1  Introduction
  1.2  a-algebras
  1.3  Measures
  1.4  Outer Measures
  1.5  Borel Measures on the Real Line
  1.6  Notes and References
2  Integration
  2.1  Measurable Functions
  2.2  Integration of Nonnegative Functions
  2.3  Integration of Complex Functions
  2.4  Modes of Convergence
  2.5  Product Measures
  2.6  The n-dimensional Lebesgue Integral
  2.7  Integration in Polar Coordinates
  2.8  Notes and References
3  Signed Measures and Differentiation
  3.1  Signed Measures
  3.2  The Lebesgue-Radon-Nikodym Theorem
  3.3  Complex Measures
  3.4  Differentiation on Euclidean Space
  3.5  Functions of Bounded Variation
  3.6  Notes and References
4  Point Set Topology
  4.1  Topological Spaces
  4.2  Continuous Maps
  4.3  Nets
  4.4  Compact Spaces
  4.5  Locally Compact Hausdorff Spaces
  4.6  Two Compactness Theorems
  4.7  The Stone-Weierstrass Theorem
  4.8  Embeddings in Cubes
  4.9  Notes and References
5  Elements of Functional Analysis
  5.1  Normed Vector Spaces
  5.2  Linear Functionals
  5.3  The Baire Category Theorem and its Consequences
  5.4  Topological Vector Spaces
  5.5  Hilbert Spaces
  5.6  Notes and References
6  LP Spaces
  6.1  Basic Theory of LP Spaces
  6.2  The Dual of LP
  6.3  Some Useful Inequalities
  6.4  Distribution Functions and Weak LP
  6.5  Interpolation of LP Spaces
  6.6  Notes and References
7  Radon Measures
  7.1  Positive Linear Functionats on Cc(X)
  7.2  Regularity and Approximation Theorems
  7.3  The Dual of Co(X)
  7.4  Products of Radon Measures
  7.5  Notes and References
8  EIementsofFourierAnalysis
  8.1  Preliminaries
  8.2  Convolutions
  8.3  The Fourier Transform
  8.4  Summation of Fourier Integrals and Series
  8.5  Pointwise Convergence of Fourier Series
  8.6  Fourier Analysis of Measures
  8. 7  Applications to Partial Differential Equations
  8.8  Notes and References
9  Elements of Distribution Theory
  9.1  Distributions
  9.2  Compactly Supported, Tempered, and Periodic
  Distributions
  9.3  Sobolev Spaces
  9.4  Notes and References
10  Topics in Probability Theory
  10.1  Basic Concepts
  10.2  The Law of Large Numbers
  10.3  The Central Limit Theorem
  10.4  Construction of Sample Spaces
  10.5  The Wiener Process
  10.6  Notes and References
11  More Measures and Integrals
  11.1  Topological Groups and Haar Measure
  11.2  Hausdorff Measure
  11.3  Self-similarity and Hausdorff Dimension
  11.4  Integration on Manifolds
  11.5  Notes and References
Bibliography
Index of Notation
Index