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

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## Elias M. Stein and Rami Shakarchi:

## 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

### 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**