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100	1		_aStein, Elias M., _d1931- _925064
245	1	0	_aReal Analysis : _bMeasure Theory, Integration, and Hilbert Spaces / _cElias M. Stein & Rami Shakarchi.
260			_aPrinceton, N.J. : _bPrinceton University Press, _c2005.
300			_axix, 402 p. : _bill. ; _c23 cm.
504			_aIncludes bibliographical references (p. 389-393) and index.
505			_aForeword vii Introduction xv 1 Fourier series: completion xvi 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 1 Preliminaries 1 The exterior measure 10 3 Measurable sets and the Lebesgue measure 16 4 Measurable functions 7 4.1 Definition and basic properties 27 4. Approximation by simple functions or step functions 30 4.3 Littlewood's three principles 33 5* The Brunn-Minkowski inequality 34 6 Exercises 37 7 Problems 46 Chapter 2: Integration Theory 49 1 The Lebesgue integral: basic properties and convergence theorems 49 2Thespace L 1 of integrable functions 68 3 Fubini's theorem 75 3.1 Statement and proof of the theorem 75 3. Applications of Fubini's theorem 80 4* A Fourier inversion formula 86 5 Exercises 89 6 Problems 95 Chapter 3: Differentiation and Integration 98 1 Differentiation of the integral 99 1.1 The Hardy-Littlewood maximal function 100 1. The Lebesgue differentiation theorem 104 Good kernels and approximations to the identity 108 3 Differentiability of functions 114 3.1 Functions of bounded variation 115 3. Absolutely continuous functions 127 3.3 Differentiability of jump functions 131 4 Rectifiable curves and the isoperimetric inequality 134 4.1* Minkowski content of a curve 136 4.2* Isoperimetric inequality 143 5 Exercises 145 6 Problems 152 Chapter 4: Hilbert Spaces: An Introduction 156 1 The Hilbert space L 2 156 Hilbert spaces 161 2.1 Orthogonality 164 2.2 Unitary mappings 168 2.3 Pre-Hilbert spaces 169 3 Fourier series and Fatou's theorem 170 3.1 Fatou's theorem 173 4 Closed subspaces and orthogonal projections 174 5 Linear transformations 180 5.1 Linear functionals and the Riesz representation theorem 181 5. Adjoints 183 5.3 Examples 185 6 Compact operators 188 7 Exercises 193 8 Problems 202 Chapter 5: Hilbert Spaces: Several Examples 207 1 The Fourier transform
520			_aReal Analysis is 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 Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.
650		0	_aFunctional analysis. _925065
650		0	_aMeasure theory. _925066
650		0	_aIntegrals, Generalized. _925067
650	1	7	_aMaattheorie. _925068
650	1	7	_aHilbertruimten. _925069
700	1		_aShakarchi, Rami. _925070
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