Hardcover ISBN:  9780821827833 
Product Code:  GSM/14.R 
List Price:  $51.00 
MAA Member Price:  $45.90 
AMS Member Price:  $40.80 
Electronic ISBN:  9781470411435 
Product Code:  GSM/14.R.E 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $38.40 

Book DetailsGraduate Studies in MathematicsVolume: 14; 2001; 348 ppMSC: Primary 28; 42; 46; 49; Secondary 26; 31;
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 minmax principle, semiclassical approximation, coherent states, LiebThirring 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 nononsense 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.ReadershipIntroductorylevel graduate students in mathematics; research mathematicians, natural scientists, and engineers interested in learning some of the important tools of modern analysis.

Table of Contents

Chapters

Chapter 1. Measure and integration

Chapter 2. $L^p$spaces

Chapter 3. Rearrangement inequalities

Chapter 4. Integral inequalities

Chapter 5. The Fourier transform

Chapter 6. Distributions

Chapter 7. The Sobolev spaces $H^1$ and $H^{1\/2}$

Chapter 8. Sobolev inequalities

Chapter 9. Potential theory and Coulomb energies

Chapter 10. Regularity of solutions of Poisson’s equation

Chapter 11. Introduction to the calculus of variations

Chapter 12. More about Eigenvalues


Additional Material

Reviews

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.
ShingTung Yau 
Begins with a downtoearth intro … aims at a wide range of essential applications … The book should work equally well in a one, or in a twosemester 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 
Praise for the previous edition … 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 twosemester 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 ... wellwritten 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


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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 minmax principle, semiclassical approximation, coherent states, LiebThirring 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 nononsense 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.
Introductorylevel graduate students in mathematics; research mathematicians, natural scientists, and engineers interested in learning some of the important tools of modern analysis.

Chapters

Chapter 1. Measure and integration

Chapter 2. $L^p$spaces

Chapter 3. Rearrangement inequalities

Chapter 4. Integral inequalities

Chapter 5. The Fourier transform

Chapter 6. Distributions

Chapter 7. The Sobolev spaces $H^1$ and $H^{1\/2}$

Chapter 8. Sobolev inequalities

Chapter 9. Potential theory and Coulomb energies

Chapter 10. Regularity of solutions of Poisson’s equation

Chapter 11. Introduction to the calculus of variations

Chapter 12. More about Eigenvalues

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.
ShingTung Yau 
Begins with a downtoearth intro … aims at a wide range of essential applications … The book should work equally well in a one, or in a twosemester 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 
Praise for the previous edition … 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 twosemester 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 ... wellwritten 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