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Main Question or Discussion Point
- Author: Elias Stein, Rami Shakarchi
- Title: Functional Analysis: Introduction to Further Topics in Analysis
- Amazon Link: https://www.amazon.com/dp/0691113874/?tag=pfamazon01-20
- Prerequisities: Real Analysis by Stein and Shakarchi
- Level: Undergrad
Table of Contents:
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[LIST]
[*] Foreword
[*] Introduction
[*] L^p Spaces and Banach Spaces
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[*] L^p spaces
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[*] The Hölder and Minkowski inequalities
[*] Completeness of Lp
[*] Further remarks
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[*] The case p = ∞
[*] Banach spaces
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[*] Examples
[*] Linear functionals and the dual of a Banach space
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[*] The dual space of L^p when 1 ≤ p < ∞
[*] More about linear functionals
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[*] Separation of convex sets
[*] The Hahn-Banach Theorem
[*] Some consequences
[*] The problem of measure
[/LIST]
[*] Complex L^p and Banach spaces
[*] Appendix: The dual of C(X)
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[*] The case of positive linear functionals
[*] The main result
[*] An extension
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[*] Exercises
[*] Problems
[/LIST]
[*] L^p Spaces in Harmonic Analysis
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[*] Early Motivations
[*] The Riesz interpolation theorem
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[*] Some examples
[/LIST]
[*] The L^p theory of the Hilbert transform
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[*] The L^2 formalism
[*] The L^p theorem
[*] Proof of Theorem 3.2
[/LIST]
[*] The maximal function and weak-type estimates
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[*] The L^p inequality
[/LIST]
[*] The Hardy space H_r^1
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[*] Atomic decomposition of H_r^1
[*] An alternative definition of H_r^1
[*] Applications to the Hilbert transform
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[*] The space H_r^1 and maximal functions
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[*] The space BMO
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Distributions: Generalized Functions
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[*] Elementary properties
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[*] Definitions
[*] Operations on distributions
[*] Supports of distributions
[*] Tempered distributions
[*] Fourier transform
[*] Distributions with point supports
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[*] Important examples of distributions
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[*] The Hilbert transform and pv(1/x)
[*] Homogeneous distributions
[*] Fundamental solutions
[*] Fundamental solution to general partial differential equations with constant coefficients
[*] Parametrices and regularity for elliptic equations
[/LIST]
[*] Calderon-Zygmond distributions and L^p estimates
[LIST]
[*] Defining properties
[*] The L^p theory
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Applications of the Baire Category Theorem
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[*] The Baire category theorem
[LIST]
[*] Continuity of the limit of a sequence of continuous functions
[*] Continuous functions that are nowhere differentiable
[/LIST]
[*] The uniform boundedness principle
[LIST]
[*] Divergence of Fourier series
[/LIST]
[*] The open mapping theorem
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[*] Decay of Fourier coefficients of L^1-functions
[/LIST]
[*] The closed graph theorem
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[*] Grothendieck's theorem on closed subspaces of L^p
[/LIST]
[*] Besicovitch sets
[*] Exercises
[*] Problems
[/LIST]
[*] Rudiments of Probability Theory
[LIST]
[*] Bernouilli trials
[LIST]
[*] Coin flips
[*] The case N=\infty
[*] Behavior of S_N as N\rightarrow \infty, first results
[*] Central limit theorem
[*] Statement and proof of the theorem
[*] Random series
[*] Random Fourier series
[*] Bernouilli trials
[/LIST]
[*] Sums of independent random variables
[LIST]
[*] Law of large numbers and ergodic theorem
[*] The role of martingales
[*] The zero-one law
[*] The central limit theorem
[*] Random variables with values in R^d
[*] Random walks
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] An Introduction to Brownian Motion
[LIST]
[*] The Framework
[*] Technical Preliminaries
[*] Construction of Brownian motion
[*] Some further properties of Brownian motion
[*] Stopping times and the strong Markov property
[LIST]
[*] Stopping times and the Blumenthal zero-one law
[*] The strong Markov property
[*] Other forms of the strong Markov Property
[/LIST]
[*] Solutions of the Dirichlet problem
[*] Exercises
[*] Problems
[/LIST]
[*] A Glimpse into Several Complex Variables
[LIST]
[*] Elementary properties
[*] Hartog's phenomenon: an example
[*] Hartog's theorem: the inhomogeneous Cauchy-Riemann equations
[*] A boundary version: the tangential Cauchy-Riemann equations
[*] The Levi form
[*] A maximum principle
[*] Approximation and extension theorems
[*] Appendix: The upper half-space
[LIST]
[*] Hardy space
[*] Cauchy integral
[*] Non-solvability
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Oscillatory Integrals in Fourier Analysis
[LIST]
[*] An illustration
[*] Oscillatory integrals
[*] Fourier transform of surface-carried measures
[*] Return to the averaging operator
[*] Restriction theorems
[LIST]
[*] Radial functions
[*] The problem
[*] The theorem
[/LIST]
[*] Application to some dispersion equations
[LIST]
[*] The Schrodinger equation
[*] Another dispersion equation
[*] The non-homogeneous Schrodinger equation
[*] A critical non-linear dispersion equation
[/LIST]
[*] A look back at the Radon transform
[LIST]
[*] A variant of the Radon transform
[*] Rotational curvature
[*] Oscillatory integrals
[*] Dyadic decomposition
[*] Almost-orthogonal sums
[*] Proof of Theorem 7.1
[/LIST]
[*] Counting lattice points
[LIST]
[*] Averages of arithmetic functions
[*] Poisson summation formula
[*] Hyperbolic measure
[*] Fourier transforms
[*] A summation formula
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Notes and References
[*] Bibliography
[*] Symbol Glossary
[*] Index
[/LIST]
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