(Ebook PDF) Functional Analysis Introduction to Further Topics in Analysis 1st Edition by Elias Stein, Rami Shakarchi ISBN 9781400840557 1400840554 full chapters

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ISBN 10:1400840554

ISBN 13:9781400840557

Author: Elias M. Stein; Rami Shakarchi

Table of Contents:

• Chapter 1. Lp Spaces and Banach Spaces
• 1 Lp spaces
• 1.1 The Hölder and Minkowski inequalities
• 1.2 Completeness of Lp
• 1.3 Further remarks
• 2 The case p = ∞
• 3 Banach spaces
• 3.1 Examples
• 3.2 Linear functionals and the dual of a Banach space
• 4 The dual space of Lp when 1 ≤ p < ∞
• 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 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 Early 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 Lp 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
• 5.1 Atomic decomposition of
• 5.2 An alternative definition of
• 5.3 Application to the Hilbert transform
• 6 The space 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
• 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 Calderón-Zygmund distributions and Lp estimates
• 3.1 Defining properties
• 3.2 The Lp 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 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 = ∞
• 1.3 Behavior of SN as N → ∞, 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 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
• 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 Hartogs’ phenomenon: an example
• 3 Hartogs’ 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 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 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

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Functional Analysis,Further Topics,Analysis,Elias Stein,Rami Shakarchi