- Paperback: 872 pages
- Publisher: Wiley-Interscience (February 1988)
- Language: English
- ISBN-10: 0471608483
- ISBN-13: 978-0471608486
- Product Dimensions: 6 x 2 x 9 inches
- Shipping Weight: 2.9 pounds (View shipping rates and policies)
- Average Customer Review: 3 customer reviews
- Amazon Best Sellers Rank: #2,224,987 in Books (See Top 100 in Books)
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Linear Operators, Part 1: General Theory (Vol 1)
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From the Publisher
This classic text, written by two notable mathematicians, constitutes a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis. Dunford and Schwartz emphasize the significance of the relationships between the abstract theory and its applications. This text has been written for the student as well as for the mathematician--treatment is relatively self-contained. This is a paperback edition of the original work, unabridged, in three volumes.
About the Author
Nelson James Dunford was an American mathematician, known for his work in functional analysis, namely integration of vector valued functions, ergodic theory, and linear operators. The Dunford decomposition, Dunford-Pettis property, and Dunford-Schwartz theorem bear his name.
Jacob Theodore "Jack" Schwartz was an American mathematician, computer scientist, and professor of computer science at the New York University Courant Institute of Mathematical Sciences. He was the designer of the SETL programming language and started the NYU Ultracomputer project.
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In some countries way off the beaten track you can
buy the answer books for these volumes. But not in America that is for sure.Unless of course you have permission.
In the days that this text was written giving students answer books in mathematics and physics was a forbidden practice. Even telling the student the answers was
unforgiveable. At least where I studied. Check out Bourbaki.
But those answer books were/are available to other students in other countries. Hence we have a kind of credibility gap here. An "unlevel playing field" if you will. More needs to be done to destroy that piracy. If I can't get the damn answer book then why should someone else be able to get it illegally? Like for example the answer book for Kelley...can you imagine it? Hell then everything would be a breeze.
Dunford and Schwartz were kind of like the advanced "Bibles" for graduate students in Mathematics in the good old days of shall we say the "Vietnam Era". When it seemed nobody including myself could keep their mind on their studies because of the politics of that period. "In the beginning God created the Real numbers...and they were perfect except that human beings couldn't count them...like the square root of two for example..."
Well that day is gone and now students can buy answer books in real analysis.
I did a review on one just last year and conspicuously the authors of that text do not mention Dunford and Schwartz at all. I wonder why.
Southern Jameson West
This book, in spite of its age, can still be thought of as a definitive treatise in the subject, even though operator theory has grown tremendously since its date of publication. There are many fascinating results in this book it is one of the few treatises on mathematics that could in some sense be thought of as respecting a certain "oral tradition" in mathematics. Indeed, the "Notes and Remarks" sections at the end of each chapter give many insights on the origins of the subject and give those who crave for a more in-depth understanding of the subject. I studied this book in great detail when I was a sophomore undergraduate, having had the privelege of doing do under the tutelage of the late functional analyst Jeffrey Butz. The excitement he generated in the course, with his lucid lectures, makes the review of this book bring back fond memories. For lack of space, only the first seven chapters will be reviewed here.
In chapter 1, the authors give a quick review of the set theory, topology, and elementary analysis that will be needed for the reader to get through the book. Functional analysis can be thought of as a generalization of linear algebra to infinite dimensions, with analysis put in so as to be able to discuss convergence. Chapter 2 then discusses the three pillars of that functional analysis is dependent on: the principle of uniform boundedness, the interior mapping principle (and its immediate corollary the closed graph theorem), and the Hahn-Banach theorem. Uniform boundedness is presented in the context of what the authors call F-spaces, and is, as they point out, a principle of equi-continuity. The interior mapping principle guarantees that continuous linear mappings bring open sets to open sets, while the Hahn-Banach theorem ensures that there are in fact functionals in the dual of a Banach space. (Banach spaces are called "B-spaces" in this book, while the dual is called the "conjugate).
Then in chapter 3, the authors take the reader through a careful treatment of measure theory and the theory of integration. They do this first via the concept of finitely additive set functions taken over a field of subsets of a set (a Boolean algebra). These set functions are taken to be the measures, and are not assumed to be bounded, and more generally than is usually the case, are assumed to have values in a real or complex Banach space. Integration theory in this set-up begins, as expected, with simple functions, and Lebesgue spaces are defined as linear spaces of measurable functions. When the set functions are countably additive, one can extend them to a wider collection, called a sigma field. This brings up the Borel field, where the set is now a topological space, and is the smallest sigma-field containing the closed sets of this space. The authors illustrate how to extend the subsequent measures via an example: the construction of the famous Radon measure on an interval. Then a metric is introduced on the sigma-field of a measure space so that it is complete, and the authors then prove the Vitali-Hahn-Saks theorem. They also show how to represent the set functions in terms of integrable functions: the Radon-Nikodym theorem.
The "special spaces" of functional analysis are overviewed in chapter 4, such as the lp and Lp spaces so familiar in applications. The authors motivate the interest in these spaces in the context of solving eight problems, dealing with how to represent the dual space, the relation between convergence in a space and its dual, weak convergence of sequences, weak completeness of a space, reflexivity of a space, weak sequential compactness of subsets of a space, compactness of subsets in the metric topology, and the connection between convergence in a space and in the space it is dual to.
Chapter 5 is an introduction to the role that convexity plays in vector spaces, wherein the authors prove an analog of the Hahn-Banach theorem, showing to what extent linear functionals can be related to convex sets. When a topology is put on the vector space, the linear functionals can determine a a topology, which for Banach spaces is the famous weak topology. This chapter is very important in applications, such as learning theory and optimization theory, due to the notions of extremal points and fixed point theorems. The Krein-Milman theorem is proven, as well as the Kakutani fixed point theorem.
The authors finally get to operator theory in chapter 6, wherein they study bounded linear maps between Banach spaces. The strong, uniform, and weak topologies are introduced immediately, adjoints are defined, along with projections. Weakly compact operators, so important in the theory of integral equations are discussed, along with the compact operators, so ubiquitous now in operator theory. Concrete examples are given on the form that operators take in various kinds of spaces, like Lebesgue spaces.
Then the reader, in chapter 7, gets totally immersed in the eigenvalue problem in infinite dimensions: the spectral theory of operators. The finite dimensional case is reviewed first, and the spectral theory of compact operators, a generalization of the Fredholm theory of linear integral equations, is discussed in great detail. This subject has changed considerably since this book was published, now being done most concisely and transparently using the language of K-theory.
However, for those of us who need applicable functional
analysis from PDE's to dynamical systems and beyond, there is no
better reference. Besides, the subject matters taken up in this
first volume have stood the test of time as the most essential
parts of functional analysis even for those primarily interested
in the subject itself. I own many books on functional analysis,
but I consider this, by far, the most useful and informative.