Analysis

Elliott H. Lieb Princeton University, Princeton, NJ, USAMichael Loss, Georgia Institute of Technology, Atlanta, GA, USA

• Preface to the First EditionPreface to the Second EditionChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12List of SymbolsReferencesIndex

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 Lp 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 JorgensenPraise 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 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