Linear Operators, General Theory (Wiley Classics Library)

Rating:
Summary: A great source of intuition.
Review: Functional analysis and operator theory is a deep and useful subject, and is still an active area of research. Quantum theory, learning theory, and probability and statistics, are just a few of the areas that make heavy use of it, and it would be difficult to formulate ideas in these areas concisely if it were not for the results that are codified in functional analysis and operator theory.

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.

Rating:
Summary: Still a great reference work
Review: Maybe it is not suitable as an introductory book on functional analysis.
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.

Rating:
Summary: Great Book for Yesteryear!
Review: There are people who love James Dean and Marilyn Monroe. This is the "Casablanca" - a great B/W movie for Functional Analysis. But, we are in a different age of big screen movie full of digital effects and sound effects. You are charged for a full price to watch an old master piece. My rating for this book is based on today's standard. Although, it is not fair for the book, but it is an unfortunate must for the reader! Of course, there are old books who stood firm in modern standard, e.g. Kolmogorov and Fermin's Introductory Real Analysis and Rudin's Functional Analysis. My rating for whether the school library should buy this book and its Part II and III is a definitely five stars!!! For beginners, my advise is don't read this intimidating thick book.