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
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