Real Analysis

by Shakarchi, Rami

• ISBN13: 9780691113869
• ISBN10: 0691113866
• Format: Hardcover
• Copyright: 2005-03-14
• Publisher: Ingram Pub Services

Summary

Real Analysisis the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series,Real Analysisis accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. Also available, the first two volumes in the Princeton Lectures in Analysis:

Table of Contents

Foreword	vii
Introduction	xv
1 Fourier series: completion	xvi
2 Limits of continuous functions	xvi
3 Length of curves	xvii
4 Differentiation and integration	xviii
5 The problem of measure	xviii
Chapter 1. Measure Theory	1	(48)
1 Preliminaries	1	(9)
2 The exterior measure	10	(6)
3 Measurable sets and the Lebesgue measure	16	(11)
4 Measurable functions	27	(7)
4.1 Definition and basic properties	27	(3)
4.2 Approximation by simple functions or step functions	30	(3)
4.3 Littlewood's three principles	33	(1)
5 The Brunn-Minkowski inequality	34	(3)
6 Exercises	37	(9)
7 Problems	46	(3)
Chapter 2. Integration Theory	49	(49)
1 The Lebesgue integral: basic properties and convergence theorems	49	(19)
2 The space L¹ of integrable functions	68	(7)
3 Fubini's theorem	75	(11)
3.1 Statement and proof of the theorem	75	(5)
3.2 Applications of Fubini's theorem	80	(6)
4 A Fourier inversion formula	86	(3)
5 Exercises	89	(6)
6 Problems	95	(3)
Chapter 3. Differentiation and Integration	98	(58)
1 Differentiation of the integral	99	(9)
1.1 The Hardy-Littlewood maximal function	100	(4)
1.2 The Lebesgue differentiation theorem	104	(4)
2 Good kernels and approximations to the identity	108	(6)
3 Differentiability of functions	114	(20)
3.1 Functions of bounded variation	115	(12)
3.2 Absolutely continuous functions	127	(4)
3.3 Differentiability of jump functions	131	(3)
4 Rectifiable curves and the isoperimetric inequality	134	(11)
4.1 Minkowski content of a curve	136	(7)
4.2 Isoperimetric inequality	143	(2)
5 Exercises	145	(7)
6 Problems	152	(4)
Chapter 4. Hilbert Spaces: An Introduction	156	(51)
1 The Hilbert space L²	156	(5)
2 Hilbert spaces	161	(9)
2.1 Orthogonality	164	(4)
2.2 Unitary mappings	168	(1)
2.3 Pre-Hilbert spaces	169	(1)
3 Fourier series and Fatou's theorem	170	(4)
3.1 Fatou's theorem	173	(1)
4 Closed subspaces and orthogonal projections	174	(6)
5 Linear transformations	180	(8)
5.1 Linear functionals and the Riesz representation theorem	181	(2)
5.2 Adjoints	183	(2)
5.3 Examples	185	(3)
6 Compact operators	188	(5)
7 Exercises	193	(9)
8 Problems	202	(5)
Chapter 5. Hilbert Spaces: Several Examples	207	(55)
1 The Fourier transform on L2	207	(6)
2 The Hardy space of the upper half-plane	213	(8)
3 Constant coefficient partial differential equations	221	(8)
3.1 Weak solutions	222	(2)
3.2 The main theorem and key estimate	224	(5)
4 The Dirichlet principle	229	(24)
4.1 Harmonic functions	234	(9)
4.2 The boundary value problem and Dirichlet's principle	243	(10)
5 Exercises	253	(6)
6 Problems	259	(3)
Chapter 6. Abstract Measure and Integration Theory	262	(61)
1 Abstract measure spaces	263	(10)
1.1 Exterior measures and Carathéodory's theorem	264	(2)
1.2 Metric exterior measures	266	(4)
1.3 The extension theorem	270	(3)
2 Integration on a measure space	273	(3)
3 Examples	276	(9)
3.1 Product measures and a general Fubini theorem	276	(3)
3.2 Integration formula for polar coordinates	279	(2)
3.3 Borel measures on R and the Lebesgue-Stieltjes integral	281	(4)
4 Absolute continuity of measures	285	(7)
4.1 Signed measures	285	(3)
4.2 Absolute continuity	288	(4)
5 Ergodic theorems	292	(14)
5.1 Mean ergodic theorem	294	(2)
5.2 Maximal ergodic theorem	296	(4)
5.3 Pointwise ergodic theorem	300	(2)
5.4 Ergodic measure-preserving transformations	302	(4)
6 Appendix: the spectral theorem	306	(6)
6.1 Statement of the theorem	306	(1)
6.2 Positive operators	307	(2)
6.3 Proof of the theorem	309	(2)
6.4 Spectrum	311	(1)
7 Exercises	312	(7)
8 Problems	319	(4)
Chapter 7. Hausdorff Measure and Fractals	323	(66)
1 Hausdorff measure	324	(5)
2 Hausdorff dimension	329	(20)
2.1 Examples	330	(11)
2.2 Self-similarity	341	(8)
3 Space-filling curves	349	(11)
3.1 Quartic intervals and dyadic squares	351	(2)
3.2 Dyadic correspondence	353	(2)
3.3 Construction of the Peano mapping	355	(5)
4 Besicovitch sets and regularity	360	(20)
4.1 The Radon transform	363	(7)
4.2 Regularity of sets when d > or equal to 3	370	(1)
4.3 Besicovitch sets have dimension 2	371	(3)
4.4 Construction of a Besicovitch set	374	(6)
5 Exercises	380	(5)
6 Problems	385	(4)
Notes and References	389	(2)
Bibliography	391	(4)
Symbol Glossary	395	(2)
Index	397