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

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