Functional Analysis

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

Foreword	p. vii
Preface	p. xvii
L^p Spaces and Banach Spaces	p. 1
L^p spaces	p. 2
The Hölder and Minkowski inequalities	p. 3
Completeness of L^p	p. 5
Further remarks	p. 7
The case p = ∞	p. 7
Banach spaces	p. 9
Examples	p. 9
Linear functionals and the dual of a Banach space	p. 11
The dual space of L^p when 1 ≤ p < ∞	p. 13
More about linear functionals	p. 16
Separation of convex sets	p. 16
The Hahn-Banach Theorem	p. 20
Some consequences	p. 21
The problem of measure	p. 23
Complex L^p and Banach spaces	p. 27
Appendix: The dual of C(X)	p. 28
The case of positive linear functionals	p. 29
The main result	p. 32
An extension	p. 33
Exercises	p. 34
Problems	p. 43
L^p Spaces in Harmonic Analysis	p. 47
Early Motivations	p. 48
The Riesz interpolation theorem	p. 52
Some examples	p. 57
The L^p theory of the Hilbert transform	p. 61
The L² formalism	p. 61
The L^p theorem	p. 64
Proof of Theorem 3.2	p. 66
The maximal function and weak-type estimates	p. 70
The L^p inequality	p. 71
The Hardy space H_r¹	p. 73
Atomic decomposition of H_r¹	p. 74
An alternative definition of H_r¹	p. 81
Application to the Hilbert transform	p. 82
The space H_r¹ and maximal functions	p. 84
The space BMO	p. 86
Exercises	p. 90
Problems	p. 94
Distributions: Generalized Functions	p. 98
Elementary properties	p. 99
Definitions	p. 100
Operations on distributions	p. 102
Supports of distributions	p. 104
Tempered distributions	p. 105
Fourier transform	p. 107
Distributions with point supports	p. 110
Important examples of distributions	p. 111
The Hilbert transform and pv(1/x)	p. 111
Homogeneous distributions	p. 115
Fundamental solutions	p. 125
Fundamental solution to general partial differential equations with constant coefficients	p. 129
Parametrices and regularity for elliptic equations	p. 131
Calderón-Zygmund distributions and L^p estimates	p. 134
Defining properties	p. 134
The L^p theory	p. 138
Exercises	p. 145
Problems	p. 153
Applications of the Baire Category Theorem	p. 157
The Baire category theorem	p. 158
Continuity of the limit of a sequence of continuous functions	p. 160
Continuous functions that are nowhere differentiable	p. 163
The uniform boundedness principle	p. 166
Divergence of Fourier series	p. 167
The open mapping theorem	p. 170
Decay of Fourier coefficients of L¹-functions	p. 173
The closed graph theorem	p. 174
Grothendieck's theorem on closed subspaces of L^p	p. 174
Besicovitch sets	p. 176
Exercises	p. 181
Problems	p. 185
Rudiments of Probability Theory	p. 188
Bernoulli trials	p. 189
Coin flips	p. 189
The case N = ∞	p. 191
Behavior of SN as N → ∞, first results	p. 194
Central limit theorem	p. 195
Statement and proof of the theorem	p. 197
Random series	p. 199
Random Fourier series	p. 202
Bernoulli trials	p. 204
Sums of independent random variables	p. 205
Law of large numbers and ergodic theorem	p. 205
The role of martingales	p. 208
The zero-one law	p. 215
The central limit theorem	p. 215
Random variables with values in Rd	p. 220
Random walks	p. 222
Exercises	p. 227
Problems	p. 235
An Introduction to Brownian Motion	p. 238
The Framework	p. 239
Technical Preliminaries	p. 241
Construction of Brownian motion	p. 246
Some further properties of Brownian motion	p. 251
Stopping times and the strong Markov property	p. 253
Stopping times and the Blumenthal zero-one law	p. 254
The strong Markov property	p. 258
Other forms of the strong Markov Property	p. 260
Solution of the Dirichlet problem	p. 264
Exercises	p. 268
Problems	p. 273
A Glimpse into Several Complex Variables	p. 276
Elementary properties	p. 276
Hartogs' phenomenon: an example	p. 280
Hartogs' theorem: the inhomogeneous Cauchy-Riemann equations	p. 283
A boundary version: the tangential Cauchy-Riemann equations	p. 288
The Levi form	p. 293
A maximum principle	p. 296
Approximation and extension theorems	p. 299
Appendix: The upper half-space	p. 307
Hardy space	p. 308
Cauchy integral	p. 311
Non-solvability	p. 313
Exercises	p. 314
Problems	p. 319
Oscillatory Integrals in Fourier Analysis	p. 321
An illustration	p. 322
Oscillatory integrals	p. 325
Fourier transform of surface-carried measures	p. 332
Return to the averaging operator	p. 337
Restriction theorems	p. 343
Radial functions	p. 343
The problem	p. 345
The theorem	p. 345
Application to some dispersion equations	p. 348
The Schrödinger equation	p. 348
Another dispersion equation	p. 352
The non-homogeneous Schrödinger equation	p. 355
A critical non-linear dispersion equation	p. 359
A look back at the Radon transform	p. 363
A variant of the Radon transform	p. 363
Rotational curvature	p. 365
Oscillatory integrals	p. 367
Dyadic decomposition	p. 370
Almost-orthogonal sums	p. 373
Proof of Theorem 7.1	p. 374
Counting lattice points	p. 376
Averages of arithmetic functions	p. 377
Poisson summation formula	p. 379
Hyperbolic measure	p. 384
Fourier transforms	p. 389
A summation formula	p. 392
Exercises	p. 398
Problems	p. 405
Notes and References	p. 409
Bibliography	p. 413
Symbol Glossary	p. 417
Index	p. 419
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