Real Analysis: Modern Techniques & Their Applications, Gerald Folland

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• Author: Gerald B. Folland
• Title: Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)
• Amazon Link: https://www.amazon.com/dp/0471317160/?tag=pfamazon01-20
• Prerequisities: Calculus, linear analysis, complex analysis, elementary set theory, linear algebra
• Level: Undergraduate, upper level; Graduate

Table of Contents:

Measures.

Integration.

Signed Measures and Differentiation.

Point Set Topology.

Elements of Functional Analysis.

L¯p Spaces.

Radon Measures.

Elements of Fourier Analysis.

Elements of Distribution Theory.

Topics in Probability Theory.

More Measures and Integrals.

Bibliography.

Indexes.

An in-depth look at real analysis and its applications-now expanded and revised.

This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.

This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.

http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471317160,descCd-description.html

GERALD B. FOLLAND is Professor of Mathematics at the University of Washington in Seattle. He has written extensively on mathematical analysis, including Fourier analysis, harmonic analysis, and differential equations.

 

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I put "lightly recommend" not because I have anything bad to say, but because I only read a tiny bit of this. What I did learn that cleared up a mystery for me was the relationship between integration in probability ("measure") and integration in differential geometry ("forms").