Functional Analysis: Introduction to Further Topics in Analysis

Functional Analysis: Introduction to Further Topics in Analysis

by Elias M. Stein, Rami Shakarchi
     
 

ISBN-10: 0691113874

ISBN-13: 9780691113876

Pub. Date: 09/11/2011

Publisher: Princeton University Press

This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors

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Overview

This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject.

  • A comprehensive and authoritative text that treats some of the main topics of modern analysis
  • A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables
  • Key results in each area discussed in relation to other areas of mathematics
  • Highlights the organic unity of large areas of analysis traditionally split into subfields
  • Interesting exercises and problems illustrate ideas
  • Clear proofs provided


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Product Details

ISBN-13:
9780691113876
Publisher:
Princeton University Press
Publication date:
09/11/2011
Pages:
442
Sales rank:
1,384,674
Product dimensions:
6.10(w) x 9.30(h) x 1.40(d)

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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