实分析(英文版)

出版时间：2013年版

内容简介

《实分析(英文)》在Princeton大学使用，同时在其它学校，比如UCLA等名校也在本科生教学中得到使用。其教学目的是，用统一的、联系的观点来把现代分析的“核心”内容教给本科生，力图使本科生的分析学课程能接上现代数学研究的脉络。

目录

ForewordIntroduction 1 Fourier series： completion 2 Limits of continuous functio 3 Length of curves 4 Differentiation and integration 5 The problem of measureChapter 1. Measure Theory 1 Prelhninaries 2 The exterior measure 3 Measurable sets and the Lebesgue measure 4 Measurable functio 4.1 Definition and basic properties 4.2 Approximation by simple functio or step functio 4.3 Littlewood's three principles 5 The Brunn-Minkowski inequality 6 Exercises 7 ProblemsChapter 2. Integration Theory 1 The Lebesgue integral： basic properties and convergencetheorems 2 The space L1 ofintegrable functio 3 Fubini's theorem 3.1 Statement and proof of the theorem 3.2 Applicatio of Fubini's theorem 4* A Fourier inveion formula 5 Exercises 6 ProblemsChapter 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 approximatio to the identity 3 Differentiability of functio 3.1 Functio of bounded variation 3.2 Absolutely continuous functio 3.3 Differentiability ofjump functio 4 Rectifiable curves and the isoperimetric inequality 4.1 Minkowski content of a curve 4.2 Isoperimetric inequality 5 Exercises 6 ProblemsChapter 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 subspaces and orthogonal projectio 5 Linear traformatio 5.1 Linear functionals and the Riesz representation theorem 5.2 Adjoints 5.3 Examples 6 Compact operato 7 Exercises 8 ProblemsChapter 5. Hilbert Spaces： Several Examples 1 The Fourier traform on L2 2 The Hardy space of the upper half-plane 3 Cotant coefficient partial differential equatio 3.1 Weaak solutio 3.2 The main theorem and key estimate 4 The Dirichlet principle 4.1 Harmonic functio 4.2 The boundary value problem and Dirichlet's principle 5 Exercises 6 ProblemsChapter 6. Abstract Measure and Integration Theory 1 Abstract measure spaces 1.1 Exterior measures and Carathodory's theorem 1.2 Metric exterior measures 1.3 The exteion theorem 2 Integration o a measure space 3 Examples 3.1 Product measures and a general Fubini theorem 3.2 Integration formula for polar coordinates 3.3 Borel measures on and the Lebesgue-Stieltjes integral 4 Absolute continuity of measures 4.1 Signed measures 4.2 Absolute continuity 5* Ergodic theorems 5.1 Mean ergodic theorem 5.2 Maximal ergodic theorem 5.3 Pointwise ergodic theorem 5.4 Ergodic measure-preserving traformatio 6* Appendix: the spectral theorem 6.1 Statement of the theorem 6.2 Positive operato 6.3 Proof of the theorem 6.4 Spectrum 7 Exercises 8 ProblemsChapter 7. Hausdorff Measure and Fractals 1 Hausdorff measure 2 Hausdorff dimeion 2.1 Examples 2.2 Self-similarity 3 Space-filling curves 3.1 Quartic intervals and dyadic squares 3.2 Dyadic correspondence 3.3 Cotruction of the Peano mapping 4* Besicovitch sets and regularity 4.1 The Radon traform 4.2 Regularity of sets when d ≥ 3 4.3 Besicovitch sets have dimeion 2 4.4 Cotruction of a Besicovitch set 5 Exercises 6 ProblemsNotes and ReferencesBibliographySymbol GlossaryIndex