Functional Analysis by Stein and Shakarchi

[micromass]

• 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:

[LIST]
[*] Foreword
[*] Introduction
[*] L^p Spaces and Banach Spaces
[LIST]
[*] L^p spaces
[LIST]
[*] The Hölder and Minkowski inequalities
[*] Completeness of Lp
[*] Further remarks
[/LIST]
[*] The case p = ∞
[*] Banach spaces
[LIST]
[*] Examples
[*] Linear functionals and the dual of a Banach space
[/LIST]
[*] The dual space of L^p when 1 ≤ p < ∞
[*] More about linear functionals
[LIST]  
[*] 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)
[LIST]
[*] The case of positive linear functionals
[*] The main result
[*] An extension
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] L^p Spaces in Harmonic Analysis
[LIST]
[*] Early Motivations
[*] The Riesz interpolation theorem
[LIST]
[*] Some examples
[/LIST]
[*] The L^p theory of the Hilbert transform
[LIST]
[*] The L^2 formalism
[*] The L^p theorem
[*] Proof of Theorem 3.2
[/LIST]
[*] The maximal function and weak-type estimates
[LIST]
[*] The L^p inequality
[/LIST]
[*] The Hardy space H_r^1
[LIST]
[*] Atomic decomposition of H_r^1
[*] An alternative definition of H_r^1
[*] Applications to the Hilbert transform
[/LIST]
[*] The space H_r^1 and maximal functions
[LIST]
[*] The space BMO
[/LIST]
[*] Exercises
[*] Problems
[/LIST]
[*] Distributions: Generalized Functions
[LIST]
[*] Elementary properties
[LIST]
[*] Definitions
[*] Operations on distributions
[*] Supports of distributions
[*] Tempered distributions
[*] Fourier transform
[*] Distributions with point supports
[/LIST]
[*] Important examples of distributions
[LIST]
[*] 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
[LIST]
[*] 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
[LIST]
[*] Decay of Fourier coefficients of L^1-functions
[/LIST]
[*] The closed graph theorem
[LIST]
[*] 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 Schrödinger equation
[*] Another dispersion equation
[*] The non-homogeneous Schrödinger 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]

 

[Figaro]

Is this book suitable to physics majors with calculus, linear algebra, differential equations background? I'm "familiar" with basic analysis i.e Abbott, but I don't do analysis every time since I'm a physics major.

 

[The Bill]

Is this book suitable to physics majors with calculus, linear algebra, differential equations background? I'm "familiar" with basic analysis i.e Abbott, but I don't do analysis every time since I'm a physics major.

No. It assumes a background in analysis including measure theory. It was originally taught as the fourth in a series of graduate level analysis courses at Princeton. The first three were somewhat independent of each other and had similar prerequisites, but this one does require material from the others or from a similar source.

This first chapter assumes the reader is familiar with measure spaces, sigma-algebras, etc.

If you want a book on basic functional analysis at the level you're asking, I recommend Applied Functional Analysis by D.H. Griffel.

 

[Figaro]

No. It assumes a background in analysis including measure theory. It was originally taught as the fourth in a series of graduate level analysis courses at Princeton. The first three were somewhat independent of each other and had similar prerequisites, but this one does require material from the others or from a similar source.

This first chapter assumes the reader is familiar with measure spaces, sigma-algebras, etc.

If you want a book on basic functional analysis at the level you're asking, I recommend Applied Functional Analysis by D.H. Griffel.

Thanks for your opinion, I've looked into Introduction to Topology and Modern Analysis by George F. Simmons, and it has chapters on basic functional analysis, as well as of course topology. What do you think about it?