Analysis (Graduate Studies in Mathematics) [Hardcover]

Elliott H. Lieb (Author), Michael Loss (Author)

Book Description

Publication Date: March 21, 2001 | ISBN-10: 0821827839 | ISBN-13: 978-0821827833 | Edition: 2

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. <P>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. <P>This new Second Edition incorporates numerous changes since the publication of the original 1997 edition and includes: <P>Features: <P>a new chapter on eigenvalues that covers the min-max principle, semi-classical approximation, coherent states, Lieb-Thirring inequalities, and more <P>extensive additions to chapters covering Sobolev Inequalities, including the Nash and Log Sobolev inequalities <P>new material on Measure and Integration <P>many new exercises <P>and much more ... <P>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, straight-forward 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. <P>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.

Editorial Reviews

Review

"Begins with a down-to-earth intro ... aims at a side 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

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

From the Publisher

This is an excellent course in operator theory and operator algebras ... leads the reader to deep new results and modern research topics ... the author has done more than just write a good book--he has managed to reveal the unspeakable charm of the subject, which is indeed the "source of happiness" for operator theorists.
-- Mathematical Reviews --This text refers to an out of print or unavailable edition of this title.

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Product Details

• Hardcover: 346 pages
• Publisher: American Mathematical Society; 2 edition (March 21, 2001)
• Language: English
• ISBN-10: 0821827839
• ISBN-13: 978-0821827833
• Product Dimensions: 10.2 x 7.2 x 1.1 inches
• Shipping Weight: 2.1 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #78,838 in Books (See Top 100 in Books)

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16 of 17 people found the following review helpful:

5.0 out of 5 stars A start in analysis., January 5, 2003

By 

Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews
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This review is from: Analysis (Graduate Studies in Mathematics) (Hardcover)

A start in analysis.-- For some number of years, Rudin's "Real and Complex", and a few other analysis books, served as the canonical choice for the book to use, and to teach from, in a first year grad analysis course. Lieb-Loss offers a refreshing alternative: It begins with a down-to-earth intro to measure theory, L^p and all that...It aims at a wide range of essential applications, such as the Fourier transform, and series, inequalities, distributions, and Sobolev spaces,--- PDE, potential theory, calculus of variations, and math physics (Schrodinger's equation, the hydrogen atom, Thomas-Fermi theory... to mention a few.) The book should work equallly well in a one, or in a two semester course. The first half of the book covers the basics, and the rest will be great for students to have, regardless of whether or not it gets to be included in a course. This choice of book is also especially agreeable to grad students in physics who need to read up on the tools of analysis.

29 of 35 people found the following review helpful:

5.0 out of 5 stars An excellent course of analysis with a theme, July 24, 1998

By 

henrique fleming - See all my reviews

This review is from: Analysis (Graduate Studies in Mathematics, V. 14) (Hardcover)

By the end of the sixties Dyson and Lennard, for the first time, proved that matter is stable. More precisely, they proved the thermodynamic stability of Coulomb matter. This was a landmark of mathematical physics, and a huge one: a very long and hard paper. A few years later, Elliott Lieb and Walter Thirring substantially improved the great Dyson result, dramatically cutting its length while improving important estimates. A very good review of these results can be find in the volume 4 of Thirring's "A Course in Mathematical Physics". Even the book version is a bit hard to read, as much mathematical analysis is required. The "Analysis" of Lieb and Loss is a book on analysis which has as a theme the great result of Lieb and Thirring. It is a real book on analysis. The chapters are named "Measure and Integration", "Lp-spaces","The Fourier transform", "Distributions", but also "Potential Theory and Coulomb En! ergies" and "Introduction to the Calculus of Variations", where nothing less than the Thomas-Fermi atom is rigorously studied. In order to leave no doubt that hard analysis is present, there are two chapters on Inequalities. After studying this splendid text the reader will be a better analist and, if he cares to, can start reading the proof of stability of matter. The proof of the pudding is NOT in the eating!

5 of 5 people found the following review helpful:

5.0 out of 5 stars A tremendous jumpstart into modern mathematical physics, May 12, 2010

By 

J. R. G. Mendonca (SP, Brazil) - See all my reviews
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This review is from: Analysis (Graduate Studies in Mathematics) (Hardcover)

I am a physicist with a somewhat limited mathematical background. However, I decided to 'break in' in modern mathematical physics, and that meant acquiring first a modicum of functional analysis and the required measure theory, harmonic analysis, and operator theory that goes with it. If you can afford the time, the classical learning path through, e.g., Kolmogorov & Fomin > Rudin (R&C) > Reed & Simon I (a path that I recommend, by the way), with possible excursions into ODEs and PDEs, probability theory, and modern geometry, is the safe way to go. If you cannot afford the time, read Lieb & Loss. It provides a tremendous jump start into what really matters for a beginning mathematical physicist: a little measure theory, L^p spaces, Sobolev spaces, a bit of Fourier analysis and PDEs, and inequalities -- lots of them (integral, Sobolev, variational). The main theorems are stated, most with proofs, and put into use.

I had a mathematical physics teacher in graduate school that once said (somewhat half-jokingly) that all you need to know is the monotone and dominated convergence theorems, the Fubini theorem, the Borel-Cantelli lemma, the Euler-Lagrange equations and how to resolve the identity using plane waves. This book by Lieb & Loss is a testimony to his confession.