**Graduate Studies in Mathematics**

Volume: 14;
2001;
348 pp;
Hardcover

MSC: Primary 28; 42; 46; 49;
Secondary 26; 31; 81

Print ISBN: 978-0-8218-2783-3

Product Code: GSM/14.R

List Price: $48.00

Individual Member Price: $38.40

**Electronic ISBN: 978-1-4704-1143-5
Product Code: GSM/14.R.E**

List Price: $48.00

Individual Member Price: $38.40

#### Supplemental Materials

# Analysis: Second Edition

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*Elliott H. Lieb; Michael Loss*

Significantly revised and expanded, this new Second Edition
provides readers at all levels—from beginning students to practicing
analysts—with the basic concepts and standard tools necessary to solve
problems of analysis, and how to apply these concepts to research in
a variety of areas.

Authors Elliott Lieb and Michael Loss take you quickly from basic
topics to methods that work successfully in mathematics and its
applications. While omitting many usual typical textbook topics,
Analysis
includes all necessary definitions, proofs, explanations, examples,
and exercises to bring the reader to an advanced level of understanding with
a minimum of fuss, and, at the same time, doing so in a rigorous and
pedagogical way. Many topics that are useful and important, but
usually left to advanced monographs, are presented in Analysis, and
these give the beginner a sense that the subject is alive and growing.

This new Second Edition incorporates numerous changes since the publication
of the original 1997 edition and includes:

Features:

- a new chapter on eigenvalues that covers the min-max principle, semi-classical approximation, coherent states, Lieb-Thirring inequalities, and more
- extensive additions to chapters covering Sobolev Inequalities, including the Nash and Log Sobolev inequalities
- new material on Measure and Integration
- many new exercises
- and much more …

The Second Edition continues its no-nonsense approach to the topic that has made it one of the best selling books on the subject. It is an authoritative, straightforward volume that readers—from the graduate student, to the professional mathematician, to the physicist or engineer using analytical methods—will find useful both as a reference and as a guide to real problem solving.

About the authors: Elliott Lieb is Professor of Mathematics and Physics at Princeton University and is a member of the US, Austrian, and Danish Academies of Science. He is also the recipient of several prizes including the 1988 AMS/SIAM Birkhoff Prize. Michael Loss is Professor of Mathematics at the Georgia Institute of Technology.

#### Readership

Introductory-level graduate students in mathematics; research mathematicians, natural scientists, and engineers interested in learning some of the important tools of modern analysis.

#### Reviews & Endorsements

This is an excellent textbook on analysis and it has several unique features: Proofs of heat kernel estimates, the Nash inequality and the logarithmic Sobolev inequality are topics that are seldom treated on the level of a textbook. Best constants in several inequalities, such as Young's inequality and the logarithmic Sobolev inequality, are also included. A thorough treatment of rearrangement inequalities and competing symmetries appears in book form for the first time. There is an extensive treatment of potential theory and its applications to quantum mechanics, which, again, is unique at this level. Uniform convexity of \(L^p\) space is treated very carefully. The presentation of this important subject is highly unusual for a textbook. All the proofs provide deep insights into the theorems. This book sets a new standard for a graduate textbook in analysis.

-- Shing-Tung Yau

Begins with a down-to-earth intro … aims at a wide range of essential applications … The book should work equally well in a one-, or in a two-semester course … great for students to have … This choice of book is also especially agreeable to grad students in physics who need to read up on the tools of analysis.

-- Palle Jorgensen

I find the selection of the material covered in the book very attractive and I recommend the book to anybody who wants to learn about classical as well as modern mathematical analysis.

-- European Mathematical Society Newsletter

The essentials of modern analysis … are presented in a rigorous and pedagogical way … readers … are guided to a level where they can read the current literature with understanding … treatment of the subject is as direct as possible.

-- Zentralblatt MATH

Lieb and Loss offer a practical presentation of real and functional analysis at the beginning graduate level … could be used as a two-semester introduction to graduate analysis … not all of the topics covered are typical. The authors introduce the subject with a thorough presentation … [an] informative exposition.

-- CHOICE

This is definitely a beautiful book … useful reference even for specialists since the authors present basic tools in a very rigorous way … they show clever methods how to calculate, equally useful for beginners as well as advanced specialists … well known exercises.

-- Mathematica Bohemica

Interesting textbook ... brings the reader quickly to a level where a wide range of topics can be appreciated ... well-written textbook ... can be read by anyone with a good knowledge of calculus ... useful for graduate students in mathematics and physics.

-- ZAMM–Journal of Applied Mathematics and Mechanics

I liked the book very much. The topics chosen were suited toward concepts that I wanted students to master.

-- Gary Sampson, Auburn University

In the area of analysis / real analysis / functional analysis there are a very large number of books at all levels, many of them very well known: the one under review is an unusual addition to the list. The book by Lieb and Loss assumes little on the part of the reader beyond a good college calculus course and, as such, begins with the basics of Lebesgue integral and yet is able to go deep into quite a few topics usually treated in advanced or more specialised texts. This unorthodox development makes it possible for a reader to reach, in the space of less than three hundred pages, completely rigorous mathematical treatment of several interesting physical problems. The authors have exercised remarkable discipline in their choice of topics to reach such depths quickly, yet they have not made it a linear development with the sole aim of showing these applications.

To sum up, this is an excellent book and the present inexpensive edition is recommended for the libraries of all interested in analysis.

-- Resonance: Journal of Science Edition

#### Table of Contents

# Table of Contents

## Analysis: Second Edition

- Cover Cover11
- Title page v6
- Contents ix10
- Preface to the first edition xvii18
- Preface to the second edition xxi22
- Chapter 1. Measure and integration 124
- Chapter 2. 𝐿^{𝑝}-spaces 4164
- Chapter 3. Rearrangement inequalities 79102
- Chapter 4. Integral inequalities 97120
- Chapter 5. The Fourier transform 125148
- Chapter 6. Distributions 137160
- Chapter 7. The Sobolev spaces 𝐻¹ and 𝐻^{1/2} 173196
- Chapter 8. Sobolev inequalities 201224
- Chapter 9. Potential theory and Coulomb energies 239262
- Chapter 10. Regularity of solutions of Poisson’s equation 259282
- Chapter 11. Introduction to the calculus of variations 269292
- Chapter 12. More about eigenvalues 301324
- List of Symbols 333356
- References 337360
- Index 343366
- Back Cover Back Cover1378