Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis, Volume IV)

✍ Scribed by Elias M. Stein, Rami Shakarchi

• Publisher: Princeton University Press
• Year: 2011
• Tongue: English
• Leaves: 442
• Series: Princeton Lectures in Analysis, 4
• Category: Library

✦ Synopsis

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

Book IV
Cover
Half-Title
Title
Copyright
Authors’ Dedications
Foreword
Contents
Preface to Book IV
Chapter 1. Lp Spaces and Banach Spaces
1.1 Lp spaces
1.1.1 The Hölder and Minkowski inequalities
1.1.2 Completeness of Lp
1.1.3 Further remarks
1.2 The case p = ∞
1.3 Banach spaces
1.3.1 Examples
1.3.2 Linear functionals and the dual of a Banach space
1.4 The dual space of Lp when 1 ≤ p < ∞
1.5 More about linear functionals
1.5.1 Separation of convex sets
1.5.2 The Hahn-Banach Theorem
1.5.3 Some consequences
1.5.4 The problem of measure
1.6 Complex Lp and Banach spaces
1.7 Appendix: The dual of C(X)
1.7.1 The case of positive linear functionals
1.7.2 The main result
1.7.3 An extension
1.8 Exercises
1.9 Problems
Chapter 2. Lp Spaces in Harmonic Analysis
2.1 Early Motivations
2.2 The Riesz interpolation theorem
2.2.1 Some examples
2.3 The Lp theory of the Hilbert transform
2.3.1 The L2 formalism
2.3.2 The Lp theorem
2.3.3 Proof of Theorem 3.2
2.4 The maximal function and weak-type estimates
2.4.1 The Lp inequality
Distribution function
2.5 The Hardy space H1 r
2.5.1 Atomic decomposition of H1
2.5.2 An alternative definition of H1
2.5.3 Application to the Hilbert transform
2.6 The space H1 r and maximal functions
2.6.1 The space BMO
2.7 Exercises
2.8 Problems
Chapter 3. Distributions: Generalized Functions
3.1 Elementary properties
3.1.1 Definitions
3.1.2 Operations on distributions
3.1.3 Supports of distributions
3.1.4 Tempered distributions
3.1.5 Fourier transform
3.1.6 Distributions with point supports
3.2 Important examples of distributions
3.2.1 The Hilbert transform and pv( 1
3.2.2 Homogeneous distributions
3.2.3 Fundamental solutions
3.2.4 Fundamental solution to general partial differential equa-
3.2.5 Parametrices and regularity for elliptic equations
3.3 Calderón-Zygmund distributions and Lp estimates
3.3.1 Defining properties
3.3.2 The Lp theory
3.4 Exercises
3.5 Problems
Chapter 4. Applications of the Baire Category Theorem
4.1 The Baire category theorem
4.1.1 Continuity of the limit of a sequence of continuous functions
4.1.2 Continuous functions that are nowhere differentiable
Proof of property (ii)
4.2 The uniform boundedness principle
4.2.1 Divergence of Fourier series
4.3 The open mapping theorem
4.3.1 Decay of Fourier coefficients of L1-functions
4.4 The closed graph theorem
4.4.1 Grothendieck’s theorem on closed subspaces of Lp
4.5 Besicovitch sets
4.6 Exercises
4.7 Problems
Chapter 5. Rudiments of Probability Theory
5.1 Bernoulli trials
5.1.1 Coin flips
5.1.2 The case N = ∞
5.1.3 Behavior of SN as N → ∞, first results
5.1.4 Central limit theorem
5.1.5 Statement and proof of the theorem
5.1.6 Random series
5.1.7 Random Fourier series
5.1.8 Bernoulli trials
5.2 Sums of independent random variables
5.2.1 Law of large numbers and ergodic theorem
5.2.2 The role of martingales
5.2.3 The zero-one law
5.2.4 The central limit theorem
5.2.5 Random variables with values in Rd
5.2.6 Random walks
5.3 Exercises
5.4 Problems
Chapter 6. An Introduction to Brownian Motion
6.1 The Framework
6.2 Technical Preliminaries
6.3 Construction of Brownian motion
6.4 Some further properties of Brownian motion
6.5 Stopping times and the strong Markov property
6.5.1 Stopping times and the Blumenthal zero-one law
6.5.2 The strong Markov property
6.5.3 Other forms of the strong Markov Property
6.6 Solution of the Dirichlet problem
6.7 Exercises
6.8 Problems
Chapter 7. A Glimpse into Several Complex Variables
7.1 Elementary properties
7.2 Hartogs’ phenomenon: an example
7.3 Hartogs’ theorem: the inhomogeneous Cauchy-Riemann equations
7.4 A boundary version: the tangential Cauchy-Riemann equations
7.5 The Levi form
7.6 A maximum principle
7.7 Approximation and extension theorems
7.8 Appendix: The upper half-space
7.8.1 Hardy space
7.8.2 Cauchy integral
7.8.3 Non-solvability
7.9 Exercises
7.10 Problems
Chapter 8. Oscillatory Integrals in Fourier Analysis
8.1 An illustration
8.2 Oscillatory integrals
8.3 Fourier transform of surface-carried measures
8.4 Return to the averaging operator
8.5 Restriction theorems
8.5.1 Radial functions
8.5.2 The problem
8.5.3 The theorem
8.6 Application to some dispersion equations
8.6.1 The Schrödinger equation
8.6.2 Another dispersion equation
8.6.3 The non-homogeneous Schrödinger equation
8.6.4 A critical non-linear dispersion equation
8.7 A look back at the Radon transform
8.7.1 A variant of the Radon transform
8.7.2 Rotational curvature
8.7.3 Oscillatory integrals
8.7.4 Dyadic decomposition
8.7.5 Almost-orthogonal sums
8.7.6 Proof of Theorem 7.1
8.8 Counting lattice points
8.8.1 Averages of arithmetic functions
8.8.2 Poisson summation formula
8.8.3 Hyperbolic measure
8.8.4 Fourier transforms
8.8.5 A summation formula
8.9 Exercises
8.10 Problems
Notes and References
Bibliography
Symbol Glossary
Index