Functional Analysis Introduction to Further Topics in Analysis 1st Edition by Elias Stein, Rami Shakarchi ISBN 978-0691113876 0691113874
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ISBN 10: 0691113874
ISBN 13: 978-0691113876
Author: Elias M. Stein, Rami Shakarchi
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:
Chapter 1. LP Spaces and Banach Spaces
1 L spaces
1.1 The Hölder and Minkowski inequalities
1.2 Completeness of LP
13 Further remarks
2 The case p = 00
3 Banach spaces
3.1 Examples
3.2 Linear functionals and the dual of a Banach space
4 The dual space of IP when 1≤ p < oc
5 More about linear functionals
5.1 Separation of convex sets
5.2 The Hahn-Banach Theorem
53 Some consequences
5.4 The problem of measure
6 Complex LP and Banach spaces
7 Appendix: The dual of 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. LP Spaces in Harmonic Analysis
1 Farly Motivations
2 The Riesz interpolation theorem
2.1 Some examples
3 The LP theory of the Hilbert transform
3.1 The L² formalism
3.2 The IP theorem
3.3 Proof of Theorem 3.2
4 The maximal function and weak-type estimates
4.1 The LP inequality
5 The Hardy space H
5.1 Atomic decomposition of H
5.2 An alternative definition of H
53 Application to the Hilbert transform
6 The space H 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 of 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 pv()
2.2 Homogeneous distributions
2.3 Fundamental solutions
2.4 Fundamental solution to gencral partial differential equations with constant coefficients
2.5 Parametrices and regularity for elliptic equations
3 Calderón-Zygmund distributions and L estimates
3.1 Defining properties
3.2 The L 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
12 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 L¹-functions
4 The closed graph theorem
4.1 Grothendieck’s theorem on closed subspaces of LP
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 = oc
1.3 Behavior of SN as Noo, first results
1.4 Central limit theorem
1.5 Statement and proof of the theorem
1.6 Random series
1.7 Random Fourier serics
1.8 Bernoulli trials
2 Sums of independent random variables
2.1 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 values in Rd
2.6 Random walks
3 Exercises
4 Problems
Chapter 6. An Introduction to Brownian Motion
1 The Frarnework
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 Hartogs’ phenomenon: an example
3 Hartogs’ theorem: the inhomogeneous Cauchy-Riemann equations
4 A boundary version: the tangential Cauchy-Riemann equa-tions
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 equations
6.1 The Schrödinger equation
6.2 Another dispersion equation
6.3 The non-homogeneous Schrödinger equation
6.4 A critical non-linear dispersion equation
7 A look back at the Radon transforın
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
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Tags: Elias Stein, Rami Shakarchi, Functional Analysis, Further Topics