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Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis) (Bk. 4), by Elias M. Stein, Rami Shakarch
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This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject.
- A comprehensive and authoritative text that treats some of the main topics of modern analysis
- A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables
- Key results in each area discussed in relation to other areas of mathematics
- Highlights the organic unity of large areas of analysis traditionally split into subfields
- Interesting exercises and problems illustrate ideas
- Clear proofs provided
- Sales Rank: #703109 in Books
- Published on: 2011-09-11
- Original language: English
- Number of items: 1
- Dimensions: 9.10" h x 1.40" w x 6.40" l, 1.65 pounds
- Binding: Hardcover
- 448 pages
Review
"Functional Analysis by Elias Stein and Rami Shakarchi is a fast-paced book on functional analysis and related topics. By page 60, you've had a decent course in functional analysis and you've got 360 pages left."--John D. Cook, Endeavour blog
"Characteristically, Stein and Shakarchi reward readers for hard work by making the material pay off."--Choice
"This excellent book ends with a proof of the continuity of the averaging operator and applications to the determination of remainder terms in asymptotic formulas for the counting function of lattice points. Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."--Stevan Pilipovic, MathSciNet, Mathematical Reviews on the Web
"This book is accessible for graduate students. Moreover, it plays the role of an instructional book in various branches of mathematical analysis, geometry, probability, and partial differential equations. In most mathematical centers one cannot expect that such lectures will be offered as a semester-long course to students, but both students and teachers have here an excellent guide for learning and teaching the topics presented in this volume. . . . Reading this book is an enjoyable experience. The reviewer highly recommends it for students and professors interested in a clear exposition of these topics."--Stevan Pilipovit, Mathematical Reviews
From the Back Cover
"This book introduces basic functional analysis, probability theory, and most importantly, aspects of modern analysis that have developed over the last half century. It is the first student-oriented textbook where all of these topics are brought together with lots of interesting exercises and problems. This is a valuable addition to the literature."--Gerald B. Folland, University of Washington
About the Author
Elias M. Stein is the Albert Baldwin Dod Professor of Mathematics at Princeton University. Rami Shakarchi received his PhD in mathematics from Princeton University. They are the coauthors of "Complex Analysis, Fourier Analysis," and "Real Analysis" (all Princeton).
Most helpful customer reviews
32 of 33 people found the following review helpful.
Professor Stein's final book is a tour de force
By Peter Haggstrom
Functional analysis is the fourth and final book in Elias Stein's and Rami Shakarchi'sPrinceton lectures in analysis. Elias Stein is a world authority on harmonic analysis and it is not surprising therefore that the first book in the series was on Fourier analysis. The second and third books covered complex and real analysis. He is also a winner of the prestigious Wolf Prize which is granted, at least in part, for excellence in communication of mathematical ideas: "For his contributions to classical and Euclidean Fourier analysis and for his exceptional impact on a new generation of analysts through his eloquent teaching and writing."
These books were based on lectures given at Princeton and therefore reflect the standard of mathematics teaching at Princeton, but more importantly they reflect what Stein wanted to do by bringing together a life time of knowledge and insights concerning Fourier theory. My understanding (which comes from someone within the functional analysis "mafia") is that Professor Stein really wanted to set a benchmark for doing Fourier series and functional analysis properly and that has informed the way he has approached these four books. The synthesis of ideas is excellent and even though I learned Fourier theory from a very able man, I was always hankering for the bigger deal - the broader connections. At the top level of mathematics the "helicopter" view is actually the hardest thing to do - just think of Littlewood's three principles: (a) every measurable set is nearly a finite sum of intervals; (b) every absolutely integrable function is nearly continuous; and (c) every pointwise convergent sequence of functions is nearly uniformly convergent. These three simple principles are but the tip of a massive analytical iceberg.
It is clear that there is a real passion for Fourier series and its tantalising applications which are extraordinarily diverse. When Fourier initially developed the seemingly outrageous theory in the early part of the 19th century, little did he know the astonishing applications that his theory would have. The sheer generality of Fourier series has in turn generated extremely subtle issues which have exercised the minds of some great mathematicians over the years. You cannot properly understand Fourier theory until you really appreciate the subtlety of the convergence issues that it poses. Stein's series of books are all about really deeply understanding why the theory works as well as it does.
Because these books are written for a mathematically sophisticated undergraduate audience they are in my view not really suitable for a struggling student. They are not suitable for an electrical engineering student, say, who just wants to know how to bang out Fourier or Laplace transforms. This is not the audience for these books. They are in fact like a complex French meal that requires a suitably chosen white or red wine to complement the overall meal. Indeed, I sometimes take one of the four volumes down to Bondi Beach to watch the waves and reflect on the depth of the material which is reinforced by the numerous exercises and problems. The exposition is very clear and the proofs are easy to follow (assuming the reader has the requisite background knowledge). There is an enormous amount of material in the exercise and problems which really amplify and reinforce the material in the text. There are some quite difficult problems but there are many hints which take you sequentially through the solution and in my experience these hints do indeed lead you systematically to the full solution. That is not to say that you don't have to do a lot of work to get there. In fact I have published detailed solutions to some of his exercises and problems.
The volume on functional analysis is actually quite different to other "classical texts" dealing with functional analysis. For instance Rudin's textbook on functional analysis has quite a different emphasis to Stein's introduction to the subject. Stein devotes a whole chapter to applications of the Baire category theory while Rudin devotes a page. Stein does this because it provides some insights into establishing the existence of a continuous but nowhere differentiable function as well as the existence of a continuous function with Fourier series diverging a point. Thus what he is doing is providing a much more holistic and integrated approach to the subject than occurs in other approaches which are much more narrowly focused. In terms of overall feel I think he is closest in philosophical approach to Frigyes Riesz whose book "Functional Analysis" (with Bela Sz.-Nagy) is so different to the more modern books. Riesz in fact "talks" through some proofs without elaborate algebra.
Stein covers the applications of functional analysis to probability theory and the vehicle he uses is Rademacher functions which enables a quick derivation of the square root law for sums of Bernoulli trials. This leads into a chapter on Brownian motion which starts with a quotation from Joe Doob which says in part that "Norbert Wiener..was so unfamiliar with the standard probability techniques even at elementary levels that his methods were so clumsily indirect that some of his own doctoral students did not realize that his Brownian motion process had independent increments". Those of us who have attempted hacking through Doob's impenetrable books will appreciate the irony in this quotation. Having said that Stein's approach to the construction of Brownian motion is different to the approaches taken by the finance world writers. He develops Brownian motion in the context of solving Dirichlet's problem generally. This is what you would expect from an expert in harmonic analysis. There is a very useful chapter oscillatory integrals in Fourier analysis which develops the theory behind averaging operators and curvature.
The book also contains all the other "usual suspects" of functional analysis - Banach spaces, LP spaces, Hardy spaces and so on.
Because this is the last book in the series it is worth going back on reviewing the scope of what has been achieved. When you do this, you appreciate what a superb job has been done in bringing the whole sprawling area together. A lifetime of work has been reflected in these books and any student who can do every single problem and exercise would indeed be destined for great things.
0 of 0 people found the following review helpful.
Five Stars
By Shalva
Very useful book in very good conditions
1 of 6 people found the following review helpful.
Excellent series!
By Al
My kid loves the Stein-Shakarchi series. He used two of their books for undergrad course work and the third as a supplement to Folland for his grad analysis course. So this last book made a perfect Xmas gift !
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