- #1
- 22,183
- 3,324
- 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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[*] 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
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[*] 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
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[*] L^p Spaces in Harmonic Analysis
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[*] Early Motivations
[*] The Riesz interpolation theorem
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[*] Some examples
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[*] 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
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[*] The maximal function and weak-type estimates
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[*] The L^p inequality
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[*] 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
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[*] Exercises
[*] Problems
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[*] 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
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[*] Calderon-Zygmond distributions and L^p estimates
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[*] Defining properties
[*] The L^p theory
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[*] Exercises
[*] Problems
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[*] Applications of the Baire Category Theorem
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[*] The Baire category theorem
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[*] Continuity of the limit of a sequence of continuous functions
[*] Continuous functions that are nowhere differentiable
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[*] The uniform boundedness principle
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[*] Divergence of Fourier series
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[*] The open mapping theorem
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[*] Decay of Fourier coefficients of L^1-functions
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[*] The closed graph theorem
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[*] Grothendieck's theorem on closed subspaces of L^p
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[*] Besicovitch sets
[*] Exercises
[*] Problems
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[*] Rudiments of Probability Theory
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[*] Bernouilli trials
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[*] 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
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[*] Sums of independent random variables
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[*] 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
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[*] Exercises
[*] Problems
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[*] An Introduction to Brownian Motion
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[*] The Framework
[*] Technical Preliminaries
[*] Construction of Brownian motion
[*] Some further properties of Brownian motion
[*] Stopping times and the strong Markov property
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[*] Stopping times and the Blumenthal zero-one law
[*] The strong Markov property
[*] Other forms of the strong Markov Property
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[*] Solutions of the Dirichlet problem
[*] Exercises
[*] Problems
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[*] A Glimpse into Several Complex Variables
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[*] 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
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[*] Hardy space
[*] Cauchy integral
[*] Non-solvability
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[*] Exercises
[*] Problems
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[*] Oscillatory Integrals in Fourier Analysis
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[*] An illustration
[*] Oscillatory integrals
[*] Fourier transform of surface-carried measures
[*] Return to the averaging operator
[*] Restriction theorems
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[*] Radial functions
[*] The problem
[*] The theorem
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[*] Application to some dispersion equations
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[*] The Schrodinger equation
[*] Another dispersion equation
[*] The non-homogeneous Schrodinger equation
[*] A critical non-linear dispersion equation
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[*] A look back at the Radon transform
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[*] A variant of the Radon transform
[*] Rotational curvature
[*] Oscillatory integrals
[*] Dyadic decomposition
[*] Almost-orthogonal sums
[*] Proof of Theorem 7.1
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[*] Counting lattice points
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[*] Averages of arithmetic functions
[*] Poisson summation formula
[*] Hyperbolic measure
[*] Fourier transforms
[*] A summation formula
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[*] Exercises
[*] Problems
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[*] Notes and References
[*] Bibliography
[*] Symbol Glossary
[*] Index
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