| 《实分析》已被哈佛大学和加利福尼亚理工学院选为教材。与《实分析》相配套的教材《傅立叶分析导论》和《复分析》也已影印出版。 |
| Stein，在国际上享有盛誉，现任美国普林斯顿大学数学系教授。 他是当代分析，特别是调和分析和分析领域领袖人物之一。古典调和分析最困难问题之一是推广到多维。他是多维欧氏调和分析的创造者之一，为此他发展了许多先进工具如奇异积分、Radon变换、极大函数等。他还发展了多个实变元的Hardy空间理论，推广了1971年F. John和L. Nirenberg的重要发现：即Hardy空间与BMO空间的对偶。在群上的调和分析方面也有贡献，例如同R.Kunze一起发现所谓Kunze-Stein现象。除此之外，他对多复变问题也做出了突出成绩。 除了研究工作之外，他的许多著作成为影响学科发展的重要参考文献。为此，他荣获1984年美国数学会在论述方面的Steele奖。 由于他的成就，他在1974年被选为美国国家科学院院士，1982年被选为美国文理学院院士，1993年获得瑞士科学院颁发的Schock奖。1999年获得世界性Wolf数学奖。 |
| foreword introduction 1 fourier series: completion 2 limits of continuous functions 3 length of curves 4 differentiation and integration 5 the problem of measure chapter 1. measure theory 1 preliminaries 2 the exterior measure 3 measurable sets and the lebesgue measure 4 measurable functions 4.1 definition and basic properties 4.2 approximation by simple functions or step functions 4.3 littlewood's three principles 5* the brunn-minkowski inequality 6 exercises 7 problems chapter 2. integration theory 1 the lebesgue integral: basic properties and convergence theorems 2 the space l1 of integrable functions 3 fubini's theorem 3.1 statement and proof of the theorem 3.2 applications of fubini's theorem 4* a fourier inversion formula 5 exercises 6 problems chapter 3. differentiation and integration 1 differentiation of the integral 1.1 the hardy-littlewood maximal function 1.2 the lebesgue differentiation theorem 2 good kernels and approximations to the identity 3 differentiability of functions 3.1 functions of bounded variation 3.2 absolutely continuous functions 3.3 differentiability of jump functions 4 rectifiable curves and the isoperimetric inequality 4.1 minkowski content of a curve 4.2* isoperimetrie inequality 5 exercises 6 problems chapter 4. hilbert spaces: an introduction 1 the hilbert space l2 2 hilbert spaces 2.1 orthogonality 2.2 unitary mappings 2.3 pre-hilbert spaces 3 fourier series and fatou's theorem 3.1 fatou's theorem 4 closed subspaees and orthogonal projections 5 linear transformations 5.1 linear flmetionals and the riesz representation the-orem 5.2 adjoints 5.3 examples 6 compact operators 7 exercises 8 problems chapter 5. hilbert spaces: several examples 1 the fourier transform on l2 2 the hardy space of the upper half-plane 3 constant coefficient partial differential equations 3.1 weak solutions 3.2 the main theorem and key estimate 4* the dirichlet principle 4.1 harmonic functions 4.2 the boundary value problem and diriehlet's principle 5 exercises 6 problems chapter 6.abstract measure and integration theory chapter 7.hausdorff measure and fractals notes and references bibliography symbol glossary index |
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