Functional Analysis: Introduction to Further Topics in Analysis

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Overview

"This book introduces basic functional analysis, probability theory, and most importantly, aspects of modern analysis that have developed over the last half century. It is the first student-oriented textbook where all of these topics are brought together with lots of interesting exercises and problems. This is a valuable addition to the literature."—Gerald B. Folland, University of Washington

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Editorial Reviews

Endeavour
Functional Analysis by Elias Stein and Rami Shakarchi is a fast-paced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left.
— John D. Cook
Choice
Characteristically, Stein and Shakarchi reward readers for hard work by making the material pay off.
MathSciNet
This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.
— Stevan Pilipovic
Mathematical Reviews
This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semester-long course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.
— Stevan Pilipovit
Endeavour - John D. Cook
Functional Analysis by Elias Stein and Rami Shakarchi is a fast-paced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left.
MathSciNet - Stevan Pilipovic
This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.
Mathematical Reviews - Stevan Pilipovit
This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semester-long course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics.
From the Publisher
"Functional Analysis by Elias Stein and Rami Shakarchi is a fast-paced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left."—John D. Cook, Endeavour blog

"Characteristically, Stein and Shakarchi reward readers for hard work by making the material pay off."—Choice

"This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."—Stevan Pilipovic, MathSciNet, Mathematical Reviews on the Web

"This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semester-long course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."—Stevan Pilipovit, Mathematical Reviews

Read More Show Less

Product Details

  • ISBN-13: 9780691113876
  • Publisher: Princeton University Press
  • Publication date: 9/11/2011
  • Pages: 442
  • Sales rank: 1,447,106
  • Product dimensions: 6.10 (w) x 9.30 (h) x 1.40 (d)

Meet the Author


Elias M. Stein is the Albert Baldwin Dod Professor of Mathematics at Princeton University. Rami Shakarchi received his PhD in mathematics from Princeton University. They are the coauthors of "Complex Analysis, Fourier Analysis", and "Real Analysis" (all Princeton).
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Table of Contents

Foreword vii

Preface xvii

Chapter 1 Lp Spaces and Banach Spaces 1

1 Lp spaces 2

1.1 The Hölder and Minkowski inequalities 3

1.2 Completeness of Lp 5

1.3 Further remarks 7

2 The case p = ∞ 7

3 Banach spaces 9

3.1 Examples 9

3.2 Linear functionals and the dual of a Banach space 11

4 The dual space of Lp when 1 ≤ p < ∞ 13

5 More about linear functionals 16

5.1 Separation of convex sets 16

5.2 The Hahn-Banach Theorem 20

5.3 Some consequences 21

5.4 The problem of measure 23

6 Complex Lp and Banach spaces 27

7 Appendix: The dual of C(X) 28

7.1 The case of positive linear functionals 29

7.2 The main result 32

7.3 An extension 33

8 Exercises 34

9 Problems 43

Chapter 2 Lp Spaces in Harmonic Analysis 47

1 Early Motivations 48

2 The Riesz interpolation theorem 52

2.1 Some examples 57

3 The Lp theory of the Hilbert transform 61

3.1 The L2 formalism 61

3.2 The Lp theorem 64

3.3 Proof of Theorem 3.2 66

4 The maximal function and weak-type estimates 70

4.1 The Lp inequality 71

5 The Hardy space Hr1 73

5.1 Atomic decomposition of Hr1 74

5.2 An alternative definition of Hr1 81

5.3 Application to the Hilbert transform 82

6 The space Hr1 and maximal functions 84

6.1 The space BMO 86

7 Exercises 90

8 Problems 94

Chapter 3 Distributions: Generalized Functions 98

1 Elementary properties 99

1.1 Definitions 100

1.2 Operations on distributions 102

1.3 Supports of distributions 104

1.4 Tempered distributions 105

1.5 Fourier transform 107

1.6 Distributions with point supports 110

2 Important examples of distributions 111

2.1 The Hilbert transform and pv(1/x) 111

2.2 Homogeneous distributions 115

2.3 Fundamental solutions 125

2.4 Fundamental solution to general partial differential equations with constant coefficients 129

2.5 Parametrices and regularity for elliptic equations 131

3 Calderón-Zygmund distributions and Lp estimates 134

3.1 Defining properties 134

3.2 The Lp theory 138

4 Exercises 145

5 Problems 153

Chapter 4 Applications of the Baire Category Theorem 157

1 The Baire category theorem 158

1.1 Continuity of the limit of a sequence of continuous functions 160

1.2 Continuous functions that are nowhere differentiable 163

2 The uniform boundedness principle 166

2.1 Divergence of Fourier series 167

3 The open mapping theorem 170

3.1 Decay of Fourier coefficients of L1-functions 173

4 The closed graph theorem 174

4.1 Grothendieck's theorem on closed subspaces of Lp 174

5 Besicovitch sets 176

6 Exercises 181

7 Problems 185

Chapter 5 Rudiments of Probability Theory 188

1 Bernoulli trials 189

1.1 Coin flips 189

1.2 The case N = ∞ 191

1.3 Behavior of SN as N → ∞, first results 194

1.4 Central limit theorem 195

1.5 Statement and proof of the theorem 197

1.6 Random series 199

1.7 Random Fourier series 202

1.8 Bernoulli trials 204

2 Sums of independent random variables 205

2.1 Law of large numbers and ergodic theorem 205

2.2 The role of martingales 208

2.3 The zero-one law 215

2.4 The central limit theorem 215

2.5 Random variables with values in Rd 220

2.6 Random walks 222

3 Exercises 227

4 Problems 235

Chapter 6 An Introduction to Brownian Motion 238

1 The Framework 239

2 Technical Preliminaries 241

3 Construction of Brownian motion 246

4 Some further properties of Brownian motion 251

5 Stopping times and the strong Markov property 253

5.1 Stopping times and the Blumenthal zero-one law 254

5.2 The strong Markov property 258

5.3 Other forms of the strong Markov Property 260

6 Solution of the Dirichlet problem 264

7 Exercises 268

8 Problems 273

Chapter 7 A Glimpse into Several Complex Variables 276

1 Elementary properties 276

2 Hartogs' phenomenon: an example 280

3 Hartogs' theorem: the inhomogeneous Cauchy-Riemann equations 283

4 A boundary version: the tangential Cauchy-Riemann equations 288

5 The Levi form 293

6 A maximum principle 296

7 Approximation and extension theorems 299

8 Appendix: The upper half-space 307

8.1 Hardy space 308

8.2 Cauchy integral 311

8.3 Non-solvability 313

9 Exercises 314

10 Problems 319

Chapter 8 Oscillatory Integrals in Fourier Analysis 321

1 An illustration 322

2 Oscillatory integrals 325

3 Fourier transform of surface-carried measures 332

4 Return to the averaging operator 337

5 Restriction theorems 343

5.1 Radial functions 343

5.2 The problem 345

5.3 The theorem 345

6 Application to some dispersion equations 348

6.1 The Schrödinger equation 348

6.2 Another dispersion equation 352

6.3 The non-homogeneous Schrödinger equation 355

6.4 A critical non-linear dispersion equation 359

7 A look back at the Radon transform 363

7.1 A variant of the Radon transform 363

7.2 Rotational curvature 365

7.3 Oscillatory integrals 367

7.4 Dyadic decomposition 370

7.5 Almost-orthogonal sums 373

7.6 Proof of Theorem 7.1 374

8 Counting lattice points 376

8.1 Averages of arithmetic functions 377

8.2 Poisson summation formula 379

8.3 Hyperbolic measure 384

8.4 Fourier transforms 389

8.5 A summation formula 392

9 Exercises 398

10 Problems 405

Notes and References 409

Bibliography 413

Symbol Glossary 417

Index 419

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