<p>Princeton Lectures in Analysis系列中的第三本，Stein的力作。</p><p>Princeton Lectures in Analysis这套丛书，是作者们为普林斯顿大学本科生写的分析学系列教材，旨在对读过大学微积分以及线性代数的学生或科研工作者，提供一套分析学的核心教材。他们依次为：(I)傅立叶分析导论，(II)复分析，(III)实分析，(IV)泛函分析。第四本还没有出，如果需要我可以把前面两本一起传上来。从2000年起，普林斯顿大学的分析学就开始使用这套教材，而现在越来越多的北美高校开始采用这套教材。</p><p>就不多说了，需要的人自然知道这本书，不知道的google一下也就知道这套教材的影响了。下面是书的目录：</p><p>Elias M. Stein &amp; Rami Shakarchi&nbsp; | 2005 | 392 pp. </p><p>TABLE OF CONTENTS:</p><p></p><p>Foreword vii<br/></p><p>Introduction xv<br/>Chapter 1. Measure Theory 1<br/>&nbsp;&nbsp;&nbsp;&nbsp; 1 Preliminaries 1<br/>&nbsp;&nbsp;&nbsp;&nbsp; 2 The exterior measure 10<br/>&nbsp;&nbsp;&nbsp;&nbsp; 3 Measurable sets and the Lebesgue measure 16<br/>&nbsp;&nbsp;&nbsp;&nbsp; 4 Measurable functions 7<br/>&nbsp;&nbsp;&nbsp;&nbsp; 5* The Brunn-Minkowski inequality 34<br/>&nbsp;&nbsp;&nbsp;&nbsp; 6 Exercises 37<br/>&nbsp;&nbsp;&nbsp;&nbsp; 7 Problems 46<br/></p><p>Chapter 2: Integration Theory 49<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 The Lebesgue integral: basic properties and convergence theorems 49<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 Thespace L 1 of integrable functions 68<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 Fubini's theorem 75<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4* A Fourier inversion formula 86<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5 Exercises 89<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6 Problems 95<br/></p><p>Chapter 3: Differentiation and Integration 98<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 Differentiation of the integral 99<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 Good kernels and approximations to the identity 108<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 Differentiability of functions 114<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4 Rectifiable curves and the isoperimetric inequality 134<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5 Exercises 145<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6 Problems 152<br/></p><p>Chapter 4: Hilbert Spaces: An Introduction 156<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 The Hilbert space L 2 156<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 Hilbert spaces 161<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 Fourier series and Fatou's theorem 170<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4 Closed subspaces and orthogonal projections 174<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5 Linear transformations 180<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6 Compact operators 188<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 7 Exercises 193<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 8 Problems 202<br/></p><p>Chapter 5: Hilbert Spaces: Several Examples 207<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 The Fourier transform on L 2 207<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 The Hardy space of the upper half-plane 13<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 Constant coefficient partial differential equations 221<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4* The Dirichlet principle 9<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5 Exercises 253<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6 Problems 259<br/></p><p>Chapter 6: Abstract Measure and Integration Theory 262<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1 Abstract measure spaces 263<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 Integration on a measure space 273<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 Examples 276<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4 Absolute continuity of measures 285<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5* Ergodic theorems 292<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6* Appendix: the spectral theorem 306<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 7 Exercises 312<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 8 Problems 319<br/></p><p>Chapter 7: Hausdorff Measure and Fractals 323<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1 Hausdorff measure 324<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2 Hausdorff dimension 329<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 3 Space-filling curves 349<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 4* Besicovitch sets and regularity 360<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 5 Exercises 380<br/>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 6 Problems 385<br/></p><p>Notes and References 389<br/>Bibliography 391<br/>Symbol Glossary 395<br/>Index 397<br/></p><p> <br/><br/></p>

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