Functional Analysis

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  • Format: Hardcover
  • Copyright: 2011-08-22
  • Publisher: Princeton Univ Pr

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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

Table of Contents

Forewordp. vii
Prefacep. xvii
Lp Spaces and Banach Spacesp. 1
Lp spacesp. 2
The Hölder and Minkowski inequalitiesp. 3
Completeness of Lpp. 5
Further remarksp. 7
The case p = ∞p. 7
Banach spacesp. 9
Examplesp. 9
Linear functionals and the dual of a Banach spacep. 11
The dual space of Lp when 1 ≤ p < ∞p. 13
More about linear functionalsp. 16
Separation of convex setsp. 16
The Hahn-Banach Theoremp. 20
Some consequencesp. 21
The problem of measurep. 23
Complex Lp and Banach spacesp. 27
Appendix: The dual of C(X)p. 28
The case of positive linear functionalsp. 29
The main resultp. 32
An extensionp. 33
Exercisesp. 34
Problemsp. 43
Lp Spaces in Harmonic Analysisp. 47
Early Motivationsp. 48
The Riesz interpolation theoremp. 52
Some examplesp. 57
The Lp theory of the Hilbert transformp. 61
The L2 formalismp. 61
The Lp theoremp. 64
Proof of Theorem 3.2p. 66
The maximal function and weak-type estimatesp. 70
The Lp inequalityp. 71
The Hardy space Hr1p. 73
Atomic decomposition of Hr1p. 74
An alternative definition of Hr1p. 81
Application to the Hilbert transformp. 82
The space Hr1 and maximal functionsp. 84
The space BMOp. 86
Exercisesp. 90
Problemsp. 94
Distributions: Generalized Functionsp. 98
Elementary propertiesp. 99
Definitionsp. 100
Operations on distributionsp. 102
Supports of distributionsp. 104
Tempered distributionsp. 105
Fourier transformp. 107
Distributions with point supportsp. 110
Important examples of distributionsp. 111
The Hilbert transform and pv(1/x)p. 111
Homogeneous distributionsp. 115
Fundamental solutionsp. 125
Fundamental solution to general partial differential equations with constant coefficientsp. 129
Parametrices and regularity for elliptic equationsp. 131
Calderón-Zygmund distributions and Lp estimatesp. 134
Defining propertiesp. 134
The Lp theoryp. 138
Exercisesp. 145
Problemsp. 153
Applications of the Baire Category Theoremp. 157
The Baire category theoremp. 158
Continuity of the limit of a sequence of continuous functionsp. 160
Continuous functions that are nowhere differentiablep. 163
The uniform boundedness principlep. 166
Divergence of Fourier seriesp. 167
The open mapping theoremp. 170
Decay of Fourier coefficients of L1-functionsp. 173
The closed graph theoremp. 174
Grothendieck's theorem on closed subspaces of Lpp. 174
Besicovitch setsp. 176
Exercisesp. 181
Problemsp. 185
Rudiments of Probability Theoryp. 188
Bernoulli trialsp. 189
Coin flipsp. 189
The case N = ∞p. 191
Behavior of SN as N → ∞, first resultsp. 194
Central limit theoremp. 195
Statement and proof of the theoremp. 197
Random seriesp. 199
Random Fourier seriesp. 202
Bernoulli trialsp. 204
Sums of independent random variablesp. 205
Law of large numbers and ergodic theoremp. 205
The role of martingalesp. 208
The zero-one lawp. 215
The central limit theoremp. 215
Random variables with values in Rdp. 220
Random walksp. 222
Exercisesp. 227
Problemsp. 235
An Introduction to Brownian Motionp. 238
The Frameworkp. 239
Technical Preliminariesp. 241
Construction of Brownian motionp. 246
Some further properties of Brownian motionp. 251
Stopping times and the strong Markov propertyp. 253
Stopping times and the Blumenthal zero-one lawp. 254
The strong Markov propertyp. 258
Other forms of the strong Markov Propertyp. 260
Solution of the Dirichlet problemp. 264
Exercisesp. 268
Problemsp. 273
A Glimpse into Several Complex Variablesp. 276
Elementary propertiesp. 276
Hartogs' phenomenon: an examplep. 280
Hartogs' theorem: the inhomogeneous Cauchy-Riemann equationsp. 283
A boundary version: the tangential Cauchy-Riemann equationsp. 288
The Levi formp. 293
A maximum principlep. 296
Approximation and extension theoremsp. 299
Appendix: The upper half-spacep. 307
Hardy spacep. 308
Cauchy integralp. 311
Non-solvabilityp. 313
Exercisesp. 314
Problemsp. 319
Oscillatory Integrals in Fourier Analysisp. 321
An illustrationp. 322
Oscillatory integralsp. 325
Fourier transform of surface-carried measuresp. 332
Return to the averaging operatorp. 337
Restriction theoremsp. 343
Radial functionsp. 343
The problemp. 345
The theoremp. 345
Application to some dispersion equationsp. 348
The Schrödinger equationp. 348
Another dispersion equationp. 352
The non-homogeneous Schrödinger equationp. 355
A critical non-linear dispersion equationp. 359
A look back at the Radon transformp. 363
A variant of the Radon transformp. 363
Rotational curvaturep. 365
Oscillatory integralsp. 367
Dyadic decompositionp. 370
Almost-orthogonal sumsp. 373
Proof of Theorem 7.1p. 374
Counting lattice pointsp. 376
Averages of arithmetic functionsp. 377
Poisson summation formulap. 379
Hyperbolic measurep. 384
Fourier transformsp. 389
A summation formulap. 392
Exercisesp. 398
Problemsp. 405
Notes and Referencesp. 409
Bibliographyp. 413
Symbol Glossaryp. 417
Indexp. 419
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