Book Description

Customer Reviews (2)

A Dover classic.
The spectral theorem of David Hilbert, John von Neumann, and Marshall Stone gives a complete answer to the question of which operators admit a diogonal representation, up to unitary equivalence, and makes the question precise as well. The theorem states that these are the normal operators in Hilbert space. This includes the selfadjoint operators which represent observables in quantum physics, and the more interesting ones are unbounded. Remember the Heisenberg commutation relations do not admit bounded solutions. But there is a mathematical distinction between formally selfadjoint operators (also called symmetric operators) and the selfadjoint ones. It is only the latter to which the spectral theorem applies. The distinction between the two is understood from a pair of indicies (n,m), now called deficiency indices. In some applications they represent
boundary conditions, and when n = m, and the boundary conditions are assigned, the symmetric operator in question has selfadjoint extensions. And we know from von Neumann what they are. A central question in the book concerns the issue of unequal indices. Then selfadjoint extensions do not exist, at least not unless the Hilbert space is enlarged. A central theme in the book is that in case of unequal indices, there is a larger Hilbert space which does in fact admit selfadjoint extensions. The co-authors, along with Naimark, are the authorities on this. Because of applications to PDE theory and to physics, there has been constant interest in the theme right up to the present. Even the current interest, and lively activity, in quantum measurement theory (in connection with quantum information theory) and entanglement brings back to to the fore this old issue around diagonalizing operators by passing to an "enlarged" (or dilated)Hilbert space, or looking for an orthonormal basis in the extended Hilbert space. So the theme of the book is still current.

Such an easy book to read!
This is a great intro to functional analysis. Having taken a graduate course on the subject, I used this as my text. The proofs are very readable and kept clear and simple. You'll see the subject develop before your eyes.One thing: when reading this book on infinite dimensional vector spaces,always try to draw a parallel with the finite dimensional version of thesubject, linear algebra. You'll appreciate the book all the more. For everytheorem relating to a bounded linear operator on Hilbert space, replace theoperator by a matrix on Euclidean n-space.. you'll say "oh yeah! Iremember that from linear algebra!" ... Read more

Book Description
This volume contains a selection of papers presented at aninternational conference on operator theory and its applicationsheld in Winnipeg. The papers chosen for this volume are intendedto illustrate that operator theory is the language of modernanalysis and its applications. Together with the papers on theabstract operator theory are many papers on the theory ofdifferential operators, boundary value problems, inversescattering and other inverse problems, and on applications tobiology, chemistry, wave propagation, and many other areas.

The volume is dedicated to the late A. V. Strauss, whoseprincipal areas of research were spectral theory of linearoperators in Hilbert spaces, extension theory for symmetriclinear operators, theory of the characteristic functions andfunctional models of linear operators, and boundary valueproblems with boundary conditions depending on spectralparameter. The bibliography of publications by A. V. Strausscombined with the papers from the conference provide bothhistorical perspective and contemporary research on the field ofoperator theory and its applications. ... Read more

Customer Reviews (1)

Nice and well organized introduction to a difficult subject
The book is a very nice introduction to the subject of Models and Decompositions of operators. In reality, it is a post-introductory book. Anyone that has already studied operator Theory in Hilbert spaces and wantsa deeper understanding of this subject, should have a look here, mainlybecause the book coherently organizes this subject that would otherwiserequire a large bibliographical research.Models and Decompositons arelooked as a sub-product of the search for the "Holy Graal" ofoperator theory: The Invariant Subspace Problem. Although it is not a bookabout that subject, it is certainly a relevant book to anyone entering thatfield, because it mentions relatively recent results that were previouslyconjectures in that area.Particularly interesting, in my opinion, arechapters 7 and 8. Chapter 7, called "Applications", in which thetheory developed in the previous chapters is applied and many interestingrelated problems, some of them still open, relative to strong and weakstability of operators are mentioned. Chapter 8 discusses the problem ofsimilarity to contractions, and leaves the reader with two related openquestions. ... Read more

Book Description
This book resulted from the lectures held at The Fields Institute(Waterloo, ON, Canada). Leading international experts presentedcurrent results on the theory of $C^*$-algebras and von Neumannalgebras, together with recent work on the classification of$C^*$-algebras. Much of the material in the book is appearinghere for the first time and is not available elsewhere in theliterature. ... Read more

Book Description
Operator theory is a diverse area of mathematics which derives its impetus and motivation from several sources. It began with the study of integral equations and now includes the study of operators and collections of operators arising in various branches of physics and mechanics. The intention of this book is to discuss certain advanced topics in operator theory and to provide the necessary background for them assuming only the standard senior-first year graduate courses in general topology, measure theory, and algebra. At the end of each chapter there are source notes which suggest additional reading along with giving some comments on who proved what and when. In addition, following each chapter is a large number of problems of varying difficulty. This new edition will appeal to a new generation of students seeking an introduction to operator theory. ... Read more

Customer Reviews (1)

Disappointing
If this book is disappointing why did I rate it 5 stars? Well, it is because having studied the first edition I was expecting more updates in the second. It really is an excellent book, but I wanted the author to discuss the Brown-Douglas-Fillmore K-theory of operator algebras and give an in-depth discussion of the invariant subspace conjecture. I used this book for a two semester course in functional analysis and operator theory while a sophomore in undergraduate and found it very challenging. The presentation of the topics is very quick and the problems are pretty difficult, but you take away an appreciation of this important area of mathematics. Indeed, the theory of Hp-spaces and Toeplitz operators, which are covered in the last two chapters, have many applications in engineering and physics. The author does give an update on the problems he marked with two stars (indicating an unsolved problem) in the first edition, and references that discuss their solution. When studying the book, I am always amazed about how rich linear transformations become when they operate on infinite-dimensional Hilbert spaces; but also how much the finite-dimensional results have generalizations in infinite dimensions. The theory of C* and W*-algebras is discussed in the book, and the presentation should be helpful to physicists who use these techniques, even though the presentation is much more general than one would find in physical theories. The explanation of the W*-algebra as arising from the enlargement of the functional calculus to an algebra of functions generated by characteristic functions; where the Gelfand transform is taken on a larger commutative self-adjoint subalgebra of the space of linear operators is one that is easier to remember and seems more natural than merely giving a set of axioms that a W*-algebra is supposed to satisfy. A very abstract discussion of index theory is given, and one that is pretty distant from the theory of integral equations, but with some work one can see the connection to these equations. Who knows, maybe a third edition with more of the author's insights? ... Read more

Book Description

Vol. 1 of Lars Hörmander's influential 4-volume treatise is a detailed exposition of the theory of distributions.

Customer Reviews (1)

the definitive guide to micro-local analysis
Hormander's four volume guide to linear partial differential operators is the definitive guide to smooth micro-local analysis from the man who invented a large fraction of it.

Micro-local analysis is an approach tothe analysis of singularities of variable coefficient linear partialdifferential equations based on the use of the Fourier transform and ofsymplectic geometry. Ideas from Hamiltonian physics and quantum theory alsofeature heavily. The fact that the subject is a mixture of analysis andgeometry makes it beautiful and powerful.

The four volume set is hardwork and in some ways is more useful as a reference work than as a learningbook but on the other hand, all the material is there and the books doreward the reader who is willing to spend the time to study themcarefully.

The first volume is the most accessible and the secondpaperback edition carries the added benefit of included exercises withpartial answers which are often highly informative. ... Read more

Book Description
In this book, Davies introduces the reader to the theory of partial differential operators, up to the spectral theorem for bounded linear operators on Banach spaces. He also describes the theory of Fourier transforms and distributions as far as is needed to analyze the spectrum of any constant coefficient partial differential operator. He also presents a completely new proof of the spectral theorem for unbounded self-adjoint operators and demonstrates its application to a variety of second order elliptic differential operators. Finally, the book contains a detailed account of the application of variational methods to estimate the eigenvalues of operators with measurable coefficients defined by the use of quadratic form techniques. Illustrated with many examples, it is well-suited to graduate-level work. ... Read more

Book Description
This book is a unique introduction to the theory of linear operators on Hilbert space. The authors' goal is to present the basic facts of functional analysis in a form suitable for engineers, scientists, and applied mathematicians. Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented. First published in 1971, Linear Operator in Engineering and Sciences has since proved to be a popular and very useful textbook. ... Read more

Customer Reviews (6)

Thorough, clear and very relevant for engineering students
At graduate school, I was required to take only 9 credit units in advanced Mathematics. Most of my credits were on Numerical Analysis, Mathematical Optimization and Linear Algebra. However, as I started covering advanced courses in dynamical systems, robust control and nonlinear systems, I found that I really needed strong foundations in Functional Analysis. I audited one course in Functional Analysis where we used the classic Rudin text. However, the pace of that class was too fast for me and I virtually learnt nothing. So I graduated with a handicap that I was unable to handle most rigorous problems. Knowing my handicap, I bought several books on this subject; I did not want to spend too much money, so each book did not cost more than $50 on Amazon marketplace. At first it was very disappointing because each book that I bought was written in the traditional format of Definition-Theorem-Proof without explaining the basic concepts. The last book to buy was this one which I read chapter by chapter from page one to the last page for about four months every evening. I believe this will be my last book on functional analysis, because it has provided what I have been looking for.

The book is well organized, and it provides detailed discussion on each topic followed with numerous and easy to understand examples that are directly related to physical systems. The reader can visualize the concepts by relating to physical systems, which is what engineering students want. After four months of evening reading, it is amazing that now I can tackle all analysis problems that used to intimidate me. This book is a must for all graduate students whose major covers dynamical systems, controls, nonlinear systems or signal processing.

Excellent applied functional analysis text
This is a great book for the applied mathematician, scientist, or engineer who wants an applicable and not overly formal introduction to functional analysis.Many examples illustrating the use of the techniques are given throughout.The text is quite readable, with a unique organization introducing algebraic, topological, and combined algebraic-topological structures and their associated theory.The book presents functional analysis without explicitly relying on the Lebesgue integral, although a well-written appendix covers Lebesgue integration and some related concepts from real variable theory.I would have appreciated more discussion of differential equations, but overall this is a very well-written book and a worthy purchase.

An eminently readable functional analysis book
An eminently readable functional analysis book including an appendix on measure theory and probability spaces. The text covers the material without explicit development of the Lebesgue integral. The notation and typesettingare very clean and the section on spectral analysis is unusually good. Theproofs are very clean and complete with numerous illustrative engineeringexamples and diagrams.I recommend this book for graduate levelengineering students.

Enlightening for mathematicians, scientists, and engineers
This is one of my favourite introductory books on functional analysis because it includes all the fundamental concepts needed to get a thorough understanding of the most important topics in this theory. It stresseswhich properties come from which structure (set-theoretic, topological, oralgebraic), and gives a complete exposition of the main results on boundedand unbounded operators, meanwhile discussing some interesting examples ofapplication. Its appendices on integration, probability and stochasticprocesses are also valuable. The text is suitable for undergraduateanalysis and functional analysis courses. Scientists and engineers willfind it very useful also because of its non-pedantic style.

Please checkmy other reviews in my member page (just click on my name above).

Continued Comment
I also recommend the "Optimizaton by vector space" by David Luenberger. It is also a very good book on linear operator with optimization applications. ... Read more

Book Description

Together with "Theory of Operator Algebras I, III" (EMS 124 and 127), this book, written by one of the most prominent researchers in the field of operator algebras, presents the theory of von Neumann algebras and non-commutative integration focusing on the group of automorphisms and the structure analysis.

Product Description

This volume contains contributions originating from the International Workshop on Operator Theory and Its Applications (IWOTA) held in Newcastle upon Tyne in July 2004. The articles expertly cover a broad range of material at the cutting edge of functional analysis and its applications. Topics include scattering and time varying systems, pseudodifferential and singular operators, weighted composition operators and hyperinvariant subspaces, and interpolation and lifting problems on Hilbert and Krein spaces.

Book Description
While there are many excellent books available on fundamental and applied electromagnetics, most introduce operator concepts in an ad hoc manner, and few discuss the subject within the general framework of operator theory. This is in contrast to quantum theory, where the use of operators and concepts from functional analysis is common. However, casting electromagnetic problems in terms of operator theory produces useful insights into the mathematical properties and physical characteristics of solutions. For instance, the commonly used modal expansion of fields in waveguides are immediately justified upon identifying the differential operators as being of the appropriate Sturm-Liouville type. As another example, existence, uniqueness and solvability of integral formulations can often be settled by appealing to the theory of Fredholm operators. Many other examples that illustrate the value of abstracting problems to an operator level are provided. Although the book focuses on mathematical fundamentals, it is written from the perspective of engineers and applied scientists working in electromagnetics. The book begins with a review of electromagnetic theory, including a discussion of singular integral operators commonly encountered in applications. It then turns to a self-contained introduction to operator theory, including basic functional analysis, linear operators, Greenand Green spectral theory, and Sturm-Liouville operators. The discussion is at an introductory mathematical level, presenting definitions and theorems, as well as proofs of the theorems when these are particularly simple or enlightening. The tools developed in this first part of the book are then applied to problems in classical electromagnetic theory: boundary-value problems and potential theory, transmission lines, waves in layered media, scattering problems in waveguides, and electromagnetic cavities. ... Read more

Customer Reviews (1)

The missing link... at last
This book represents what I personally think is the missing link. Which link is that and between what?... well, the book introduces the reader to the very useful and easy, yet rigors theory of operators. This analytical tool is outstanding when someone wants to start proper analysis of an electromagnetic problem and wishes to do it in higher level of abstraction on the paper yet, with high level of details and physical understanding in mind. Therefore, The book is the link between the electromagnetism and theory of operators. The book is very useful for engineers and physicists who are not particularly interested in pure mathematics but instead, interested in the theory of operators as a powerful tool for both analytical and numerical formulations in electromagnetics. As a result, this book for advanced users and I suggest the beginners and intermediate readers to start with the book by Donald G. Dudley "Mathematical Foundations for Electromagnetic Theory ". Other sources that are of a great importance and support this book and make it perfect is the two by Ismo V. Lindell "Methods for Electromagnetic Field Analysis" and "Differential forms in Ellectromagnetics" ... Read more

Product Description
Operator theory is a significant part of many important areas of modern mathematics: functional analysis, differential equations, index theory, representation theory, mathematical physics, and more. This text covers the central themes of operator theory, presented with the excellent clarity and style that readers have come to associate with Conway's writing.Early chapters introduce and review material on C*-algebras, normal operators, compact operators and non-normal operators. The topics include the spectral theorem, the functional calculus and the Fredholm index. Also, some deep connections between operator theory and analytic functions are presented.Later chapters cover more advanced topics, such as representations of C*-algebras, compact perturbations and von Neumann algebras. Major results, such as the Sz.-Nagy Dilation Theorem, the Weyl-von Neumann-Berg Theorem and the classification of von Neumann algebras, are covered, as is a treatment of Fredholm theory. These advanced topics are at the heart of current research.The last chapter gives an introduction to reflexive subspaces, i.e., subspaces of operators that are determined by their invariant subspaces. These, along with hyperreflexive spaces, are one of the more successful episodes in the modern study of asymmetric algebras.Professor Conway's authoritative treatment makes this a compelling and rigorous course text, suitable for graduate students who have had a standard course in functional analysis. ... Read more

Book Description
Basic Operator Theory provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential and integral equations, approximation theory, and numerical analysis. A textbook designed for senior undergraduate and graduate students, Basic Operator Theory begins with the geometry of Hilbert space and proceeds to the spectral theory for compact self-adjoint operators with a wide range of applications. Part of the volume is devoted to Banach spaces and operators acting on these spaces.Presented as a natural continuation of linear algebra, Basic Operator Theory provides a firm foundation in operator theory, an essential part of mathematical training for students of mathematics, engineering, and other technical sciences. ... Read more

Customer Reviews (3)

A great source of intuition.
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 definedas 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 problemin 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.

Still a great reference work
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.

Great Book for Yesteryear!
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 andsound effects. You are charged for a full price to watch an old masterpiece. My rating for this book is based on today's standard. Although, itis not fair for the book, but it is an unfortunate must for the reader! Ofcourse, there are old books who stood firm in modern standard, e.g.Kolmogorov and Fermin's Introductory Real Analysis and Rudin's FunctionalAnalysis. My rating for whether the school library should buy this book andits Part II and III is a definitely five stars!!! For beginners, my adviseis don't read this intimidating thick book. ... Read more

Book Description
This is one of the few books available in the literature that contains problems devoted entirely to the theory of operators on Banach spaces and Banach lattices. The book contains complete solutions to the more than 600 exercises in the companion volume, An Invitation to Operator Theory, Volume 50 in the AMS series Graduate Studies in Mathematics, also by Abramovich and Aliprantis.

The exercises and solutions contained in this volume serve many purposes. First, they provide an opportunity to the readers to test their understanding of the theory. Second, they are used to demonstrate explicitly technical details in the proofs of many results in operator theory, providing the reader with rigorous and complete accounts of such details. Third, the exercises include many well-known results whose proofs are not readily available elsewhere. Finally, the book contains a considerable amount of additional material and further developments. By adding extra material to many exercises, the authors have managed to keep the presentation as self-contained as possible.

The book can be very useful as a supplementary text to graduate courses in operator theory, real analysis, function theory, integration theory, measure theory, and functional analysis. It will also make a nice reference tool for researchers in physics, engineering, economics, and finance. ... Read more

Book Description

This work treats quantitative aspects of the approximation of functions using positive linear operators. The theory of these operators has been an important area of research in the last few decades, particularly as it affects computer-aided geometric design. In this book, the crucial role of the second order moduli of continuity in the study of such operators is emphasized. New and efficient methods, applicable to general operators and to diverse concrete moduli, are presented. The advantages of these methods consist in obtaining improved and even optimal estimates, as well as in broadening the applicability of the results.

Book Description
This book is aimed at a new generation of researchers and graduate students in the field. Written in a user-friendly, motivating style, fundamental topics are presented in a systematic fashion, i.e., set theory, algebraic structures, topological structures, Banach spaces, Hilbert Spaces, culminating with the Spectral Theorem, one of the landmarks in the theory of operators on Hilbert spaces. The exposition is concept-driven and as much as possible avoids the formula-computational approach. . Key features of this largely self-contained work include: * required background material to each chapter * fully rigorous proofs, over 300 of them, are specially tailored to the presentation and some are new * more than 100 examples and, in several cases, interesting counterexamples that demonstrate the frontiers of an important theorem * over 300 problems, many with hints * both problems and examples underscore further auxiliary results and extensions of the main theory; in this non-traditional framework, the reader is challenged and has a chance to prove the principal theorems anew. This work is an excellent text for the classroom as well as a self-study resource for researchers. Prerequisites include an introduction to analysis and to functions of a complex variable, which most first-year graduate students in mathematics, engineering, or another formal science have already acquired. ... Read more

Customer Reviews (4)

Good introduction
I enjoyed reading this book. As another reviewer said, the proofs are a little too detailed. However, for my purposes, a little hand-holding was just what I needed. I haven't read any other books on the subject, so I can't make a thorough comparison. However, I have been reading sections of Conway's "A Course In Functional Analysis". Here are some surface differences.

1. Both books have a large number of typos and other errata. Both authors provide an errata page on the web. Conway's book is in it's third printing so as expected, it has fewer errata. In any case, most of the errata are typos that are easy to spot and fix on your own.

2. The proofs in Kubrusly's book are far more detailed than Conway's. As a result, you can read it much faster, but you don't get as much out of it.

3. Conway covers much more ground than Kubrusly. Some areas that I am interested in, Banach Algebras, the Riesz functional calculus, and unbounded operators are not covered at all in Kubrusly.

I have this complaint about Professor Kubrusly's proofs. It is not always clear exactly what he is proving. If the theorem is of the 'if and only if' type, he will start his proof without telling you which half he is doing. In the proof of the Arzela-Ascoli theorem, he doesn't tell you what he is proving until two pages after he starts the proof. I suppose this is not such a big deal since eventually you do find out what he is proving.

For someone like me who wants to go deeper into the field, this book is a good introduction, but requires you to read a second book. For someone who really only needs the material that is covered in the book, you will find it a painless introduction.

A useful book, great for the beginner
This is a very self-contained book on fundamentals of functional analysis. The pace is very slow and the proofs too detailed at times. This is particularly so in Chapter 5 on Hilbert Spaces. The good thing about this book are the problems at the end of each chapter. They are mostly standard problems and a lot of 'hints' are provided (as opposed to most books in this field). I have a number of books on functional analysis but often I go back to Kubrusly's for some fundamental notion. One downside is the brief treatment on the spectral theorem for normal operators, but I guess the author's intention was too avoid using measure theoretic results, which most engineering students are not familiar with. A great pick for the engineer seeking a better understanding of higher mathematics.

Simply Indispensable
This is a must have Analisys book. Although the modest title, the book is the best introduction to functional analysis ever written. More than this, its an indispensable first book for everyone interested in operator theory. It is clear, concise and all the exercises are really solvable. You feel that the autor wants to teach you. It requires only a modest (really!) background in linear algebra and calculus. At the end, you will have learned all real analysis that so many autors (and teachers!) always have hidden from you. There are so many math disclosure in the book that I'm afraid Kubrusly being at risk of attack from any xiita math (especially the mediocre one). I agree with the last reviewer, the book is an immediate classic and Kubrusly has turned himself into a kind of Salman Rushdie for his daring in writing the Satanic Verses of the Mathematics.

It's a imediate classic!!
For those wanting to learn not only functional analisys but also the basic elements about math (set thoery, algebra, topology,...), I doubt that they can find anything better than this book. It's the state of art!!! ... Read more