# Book:Elias M. Stein/Functional Analysis: An Introduction to Further Topics in Analysis

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## Elias M. Stein and Rami Shakarchi: Functional Analysis: An Introduction to Further Topics in Analysis

Published $\text {2011}$, Princeton University Press

ISBN 978-0691113876.

### Subject Matter

Functional Analysis

### Contents

Foreword

Preface

Chapter 1. $L^p$ Spaces and Banach Spaces
1. $L^p$ spaces
1.1 The Holder and Minkowski inequalities
1.2 Completeness of $L^p$
1.3 Further remarks
2. The case $p = \infty$
3. Banach spaces
3.1 Examples
3.2 Linear functionals and the dual of a Banach space
4. The dual space of $L^p$ when $1 \le p < \infty$
5. More about linear functionals
5.1 Separation of convex sets
5.2 The Hahn-Banach Theorem
5.3 Some consequences
5.4 The problem of measure
6. Complex $L^p$ and Banach spaces
7. Appendix: The dual of $\map C X$
7.1 The case of positive linear functionals
7.2 The main result
7.3 An extension
8. Exercises
9. Problems
Chapter 2. $L^p$ Spaces in Harmonic Analysis
1. Early motivations
2. The Riesz interpolation theorem
2.1 Some examples
3. The $L^p$ theory of the Hilbert transform
3.1 The $L^2$ formalism
3.2 The $L^p$ theorem
3.3 Proof of Theorem $3.2$
4. The maximal function and weak-type estimates
4.1 The $L^p$ inequality
5. The Hardy space $H^1_r$
5.1 Atomic decomposition of $H^1_r$
5.2 An alternative definition of $H^1_r$
5.3 Application to the Hilbert transform
6. The space $H^1_r$ and maximal functions
6.1 The space BMO
7. Exercises
8. Problems
Chapter 3. Distributions. Generalized Functions
1. Elementary properties
1.1 Definitions
1.2 Operations on distributions
1.3 Supports on distributions
1.4 Tempered distributions
1.5 Fourier transform
1.6 Distributions with point supports
2. Important examples of distributions
2.1 The Hilbert transform and $\map {\text {pv} } {\frac 1 x}$
2.2 Homogeneous distributions
2.3 Fundamental solutions
2.4 Fundamental solution to general partial differential equations with constant coefficients
2.5 Parametrices and regularity for elliptic equations
3. Calderon-Zygmund distributions and $L^p$ estimates
3.1 Defining properties
3.2 The $L^p$ theory
4. Exercises
5. Problems
Chapter 4. Applications of the Baire Category Theorem
1. The Baire category theorem
1.1 Continuity of the limit of a sequence of continuous functions
1.2 Continuous functions that are nowhere differentiable
2. The uniform boundedness principle
2.1 Divergence of Fourier series
3. The open mapping theorem
3.1 Decay of Fourier coefficients of $L^1$ functions
4. The closed graph theorem
4.1 Grothendieck's theorem on closed subspaces of $L^p$
5. Besicovitch sets
6. Exercises
7. Problems
Chapter 5. Rudiments of Probability Theory
1. Bernoulli trials
1.1 Coin flips
1.2 The case $N = \infty$
1.3 Behaviour of $S_N$ as $N \rightarrow \infty$, first results
1.4 Central limit theorem
1.5 Statement and proof of the theorem
1.6 Random series
1.7 Random Fourier series
1.8 Bernoulli trials
2. Sums of independent random variables
2.1 The law of large numbers and ergodic theorem
2.2 The role of martingales
2.3 The zero-one law
2.4 The central limit theorem
2.5 Random variables with $\R^d$
2.6 Random walks
3. Exercises
4. Problems
Chapter 6. An Introduction to Brownian Motion
1. The Framework
2. Technical preliminaries
3. Construction of Brownian motion
4. Some further properties of Brownian motion
5. Stopping times and the strong Markov property
5.1 Stopping times and the Blumenthal zero-one law
5.2 The strong Markov property
5.3 Other forms of the strong Markov property
6. Solution of the Dirichlet problem
7. Exercises
8. Problems
Chapter 7. A Glimpse into Several Complex Variables
1. Elementary properties
2. Hartog's phenomenon: an example
3. Hartog's theorem: the inhomogeneous Cauchy-Riemann equations
4. A boundary version: the tangential Cauchy-Riemann equations
5. The Levi form
6. A maximum principle
7. Approximation and extension theorems
8. Appendix: The upper half-space
8.1 Hardy space
8.2 Cauchy integral
8.3 Non-solvability
9. Exercises
10. Problems
Chapter 8. Oscillatory Integrals in Fourier Analysis
1. An illustration
2. Oscillatory integrals
3. Fourier transform of surface-carried measures
4. Return to the averaging operator
5. Restriction theorems
5.1 Radial functions
5.2 The problem
5.3 The theorem
6. Application to some dispersion theorems
6.1 The Schrödinger equation
6.2 Another dispersion relation
6.3 The non-homogeneous Schrödinger equation
6.4 A critical non-linear dispersion relation
7. A look back at the Radon transform
7.1 A variant of the Radon transform
7.2 Rotational curvature
7.3 Oscillatory integrals
7.4 Dyadic decomposition
7.5 Almost-orthogonal sums
7.6 Proof of theorem $7.1$
8. Counting lattice points
8.1 Averages of arithmetic functions
8.2 Poisson summation formula
8.3 Hyperbolic measure
8.4 Fourier transforms
8.5 A summation formula
9. Exercises
10. Problems

Notes and References

Bibliography

Symbol Glossary

Index