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This is the second part of an elementary textbook which combines linear functional analysis, nonlinear functional analysis, and their substantial applications with each other. The book addresses undergraduate students and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems which relate to our real world and which play an important role in the history of mathematics. The book's approach begins with the question "what are the most important applications" and proceeds to try to answer this question. The applications concern integral equations, differential equations, bifurcation theory, the moment problem, Cebysev approximation, the optimal control of rockets, game theory, symmetries and conservation laws (the Noether theorem), the quark model, and gauge theory in elementary particle physics. The presentation is self-contained. As for prerequisites, the reader should be familiar with some basic facts of calculus.

The first part of this textbook has been published under the title Applied Functional Analysis: Applications to Mathematical Physics. ... Read more

Customer Reviews (1)

A perspectiveful book
This excellent volume is exactly what you need to get a fast connection between traditional abstract operator theory and other application-oriented branches of mathematics. It shows how apparently different topics are in fact closely related. For example: The Hahn-Banach theorem and separationof convex sets, and eventually the optimal control of rockets; or reflexiveBanach spaces and obstacle problems in elasticity; or Fredholm operatorsand the Navier-Stokes equations.

If find this text suitable mostly forgraduate students who are already acquainted with this kind of subjects andwant to deepen their knowledge on the interaction between differentbranches of mathematics and physics.

Please check my other reviews in mymember page (just click on my name above). ... Read more

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The Colloquium series bestseller. ... Read more

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Clinicians and students of clinical psychology and behavior therapy will welcome this overview of the important process of functional analysis, particularly the major developments in methods and practice over the last fifteen years. Recent years have seen new questionnaires, interview formats, observational methods and strategies for both the practitioner and the researcher. This book arose principally from a course for clinical psychology students working with a wide range of different clinical populations including adults, children, adolescents, families, older adults and persons with long-term mental health disabilities. The methods and experience reviewed here will be useful to clinical psychologists and behavior therapists working in any problem field. This book appears in The Wiley Series in Clinical Psychology Series Editor: J. Mark G. Williams University of Wales, Bangor, UK ... Read more

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This book was written to address the need for timely, thorough, practical, and defensible job analysis for HR managers. Under continuing development over the past 50 years, Functional Job Analysis (FJA) is acknowledged by major texts in HR and industrial/organizational psychology as one of the premier methods of job analysis used by leading-edge organizations in the private and public sectors. It is unique among job analysis methods in having its own in-depth theoretical grounding within a systems framework. In addition to providing a methodology for analyzing jobs, it offers a rich model and vocabulary for communicating about the competencies (skills) contributing to work success and about the design of the work organization through which those competencies are expressed.

FJA is the right theory and methodology for future work in an increasingly competitive global economy. This book is the authoritative source describing how FJA can encourage and support an ongoing dialogue between workers and management as they jointly pursue total quality, worker growth, and organization performance. It is a flexible tool, fully recognizing the rapid changes impacting today's organizations. It is a comprehensive tool, leading to an in-depth understanding of work, its results, and its improvement in a unique organization context. It is a humane tool, viewing workers in light of their full potential and capacity for positive growth. With FJA, workers and managers can work more constructively together in a wholesome and productive work relationship.
... Read more

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The present book is a self-contained text which leads the reader through all the important aspects of the theory of locally convex vector spaces over nonarchimedean fields. One can observe an increasing interest in methods from nonarchimedean functional analysis, particularly in number theory and in the representation theory of p-adic reductive groups. The book gives a concise and clear account of this theory, it carefully lays the foundations and also develops the more advanced topics. Although the book will be a valuable reference work for experts in the field, it is mainly intended as a streamlined but detailed introduction for researchers and graduate students who wish to apply these methods in different areas. ... Read more

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Massive compilation offers balanced approach between theory and application. Detailed, in-depth discussions of vector spaces and topological vector spaces, the Hahn-Banach Theorem, fixed-point theorems, duality theory, theory of compact operators, Krein-Milman Theorem and its applications, much more. Many examples and a large number of exercises are included, along with a comprehensive 32-page bibliography. 1965 edition.
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Highly recommended
This is the fourth of E. Zeidler's amazing 5 volume set (I,IIa,IIb,III,IV)on nonlinear functional analysis and it's applications. I am a real fan ofthese books and have used this particularvolume to understand variousfine points of variational theory. For example, I very much liked chapter48 on generalized Lagrangemultipliers (Dubovickii-Miljutin theory).

See my review of volume 1 for more extensive comments. ... Read more

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Customer Reviews (2)

Kudos to Zimmer on another hit
As the previous reviewer so eloquently put it, this book is quite simply the best there is on functional analysis. It explains beautifully the incredibly complex and diverse topic of functional analysis and sheds light on many of the most mysterious complexities in the field. We all owe Robert Zimmer our appreciation for this hit

Exelente!
Creo que este es una de las revisiones mas completas sobre Analisis Funcional que he podido leer , creo da un espectro completo de el Analisis Funcional, el modo en que se encara cada capitulo tiene un estilo muyparticular, en especial los ejemplos son tan ilustrativos que forman parteesencial del libro, considero este un trabajo realmente bueno, como pocosen esta tan fructifera area... ... Read more

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This is a graduate text on functional analysis. After presenting the fundamental function spaces and their duals, the authors study topics in operator theory and finally develop the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems. Along the way, the reader is presented with a truly remarkable assortment of well formulated and interesting exercises, which test the understanding as well as point out many related topics. The answers and hints that are not already contained in the statements of the exercises are collected at the end of the book. ... Read more

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Analyzes the theory of normed linear spaces and of linear mappings between such spaces, providing the necessary foundation for further study in many areas of analysis. Strives to generate an appreciation for the unifying power of the abstract linear-space point of view in surveying the problems of linear algebra, classical analysis, and differential and integral equations. This second edition incorporates recent developments in functional analysis to make the selection of topics more appropriate for current courses in functional analysis. Additions to this new edition include: a chapter on Banach algebras, and material on weak topologies and duality, equicontinuity, the Krein-Milman theorem, and the theory of Fredholm operators. Greater emphasis is also placed on closed unbounded linear operators, with more illustrations drawn from ordinary differential equations. ... Read more

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Based on a third-year course for French students of physics, this book is a graduate text in functional analysis emphasizing applications to physics. It introduces Lebesgue integration, Fourier and Laplace transforms, Hilbert space theory, theory of distribution a la Laurent Schwartz, linear operators, and spectral theory. It contains numerous examples and completely worked out exercises. ... Read more

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Functional analysis plays a crucial role in the applied sciences as well as in mathematics. It is a beautiful subject that can be motivated and studied for its own sake. In keeping with this basic philosophy, the author has made this introductory text accessible to a wide spectrum of students, including beginning-level graduates and advanced undergraduates. The exposition is inviting, following threads of ideas, describing each as fully as possible, before moving on to a new topic. Supporting material is introduced as appropriate, and only to the degree needed. Some topics are treated more than once, according to the different contexts in which they arise. The prerequisites are minimal, requiring little more than advanced calculus and no measure theory. The text focuses on normed vector spaces and their important examples, Banach spaces and Hilbert spaces. The author also includes topics not usually found in texts on the subject. This Second Edition incorporates many new developments while not overshadowing the book's original flavor. Areas in the book that demonstrate its unique character have been strengthened. In particular, new material concerning Fredholm and semi-Fredholm operators is introduced, requiring minimal effort as the necessary machinery was already in place. Several new topics are presented, but relate to only those concepts and methods emanating from other parts of the book. These topics include perturbation classes, measures of noncompactness, strictly singular operators, and operator constants. Overall, the presentation has been refined, clarified, and simplified, and many new problems have been added. ... Read more

Customer Reviews (1)

A good start
Functional analysis is now a vast subject with many applications and this book gives an introduction to this branch of mathematics that can be understood by advanced undergraduates or beginning graduate students. The book fills a gap between elementary aspects of analysis and those using measure theory. The author motivates the subject matter very well, and therefore gives the reader a deeper appreciation of the basics of functional analysis and operator theory. Because of its high quality of presentation, it is very unfortunate therefore that the book is out of print.

In chapter 1, the author introduces the subject by considering a particular differential equation, which he solves by the method of variation of parameters. Generalizing the equation leads him finally to the Volterra integral equation, and by abstracting from the properties of this equation he introduces the notion of a Banach space. Several examples of these spaces are given, and he uses them to introduce Hilbert spaces. Appropriately, a discussion of Fourier series concludes the chapter.

The Riesz representation theorem, which shows that every bounded linear functional on a Hilbert space can be represented as an inner product, is introduced immediately in chapter 2. This representation is used to motivate a similar question for Banach spaces, namely whether Banach spaces have any nonzero bounded linear functionals. The Hahn-Banach theorem, discussed in detail, shows there are many such functionals. Since there are so many, it makes sense to speak of the space of all of them, called the dual space, which is a Banach space even though the original space may not be. Several examples of dual spaces are given.

The author discusses the space of bounded linear operators between normed linear spaces X and Y in chapter 3, showing that it is a Banach space ifY is. The adjoint of an operator is defined, and the author shows why adjoints are useful. The famous Closed Graph Theorem and its geometric interpretation are discussed in detail. Another fundamental result, the Uniform Boundedness Theorem, is proven.

Integral equations are also used to motivate the concept of a compact operator in chapter 4, but first via the operators of finite rank. The compact operators are thus the limit in norm of operators of finite rank, and their adjoints are also compact.

One of the most important classes of operators of all, the Fredholm operators, are discussed in chapter 5. The author discusses what can be concluded from their definition and gives other criteria for recognizing when an operator is Fredholm. The index theory of Fredholm operators is covered in this chapter, and this is a theory that has far-reaching implications in other areas of mathematics, such as algebraic topology and K-theory. The author discussed briefly a collection of operators called semi-Fredholm operators, that are defined by relaxing some of the conditions for Fredholm operators are relaxed.

Spectral theory, also of enormous importance in operator theory, is discussed in chapter 6. The author proves the spectral mapping theorem, explaining the need for complex Banach spaces in this regard. The author shows how to obtain a formula for the inverse of a bounded operator polynomial. The reader will need a background in complex variables in order to follow his discussion here, which has become known as the "operational calculus". If the spectrum of the operator is disconnected, the author shows how to obtain the spectral projections of the operator.

The theory of unbounded operators, typically not treated in beginning books on operator theory, but very important in applications, is discussed in chapter 7. Unbounded Fredholm operators are defined, and the author shows just how much the bounded results can be carried over to these. The same is done for unbounded semi-Fredholm operators.

The author returns to Banach spaces in chapter 8, where he discusses reflexive Banach spaces. He proves that a Banach space is reflexive if its dual is, and shows how "saturated subspaces" characterize reflexive spaces. The very important weak* convergence appears here in the discussion of separable Banach spaces.

Chapter 9 is the beginning of "soft analysis" wherein the author discusses Banach algebras. The reader gets his first taste of the intertwining of operator theory, algebra, and analysis in this chapter. The "soft" approach to operator theory is vast and is marked by many brilliant developments.

Semigroups, so very important in physical applications, are discussed in chatper 10. The discussion again motivated by a differential equation, the author shows clearly the need for strong continuity in obtaining an infinitesimal generator for a one-parameter semigroup of operators.

The author returns to Hilbert spaces in chapter 11, wherein he introduces the highly important class of normal operators. He shows that such operators are norm attaining, and proves when normal compact operators have a complete, orthonormal set of eigenvectors. He applies these considerations to a class of (compact) integral operators and then discusses briefly the theory of hyponormal operators.

Another topic not usually included in treatments at this level is the theory of bilinear forms, which is done in chapter 12. Very important in many areas of mathematics, the author shows their utility in the context of dissipative operators. He then goes on to the topic of self-adjoint extensions, showing that every densely defined symmetric linear operator has a self-adjoint extension.

Self-adjoint opertaors are then treated in detail in chapter 13, beginning with the theory of orthogonal projections, which are very important in applications like quantum physics. He then proves the spectral resolution theorem for bounded self-adjoint operators.

The last chapter of the book details some applications, the first few sections dealing mainly with a particular (unbounded) differential operator. Using two different methods the author calculates the spectrum of this operator. He then does essentially the same analysis for a simple integral operator. Both of these operators appear many times in physical applications. ... Read more

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This nicely written manuscript takes a gentler approach than other functional analysis graduate texts, and includes an improved approach along with a better choice of topics. The concise treatment makes this ideal for a one-semester course. The exercises in this manuscript are numerous and of a very high quality. Interesting historical tidbits are scattered throughout the text, many of which will be new to most readers. The main prerequisites are basic undergraduate courses in real analysis, linear algebra, and point set topology.

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In an age where the amount of data collected from brain imaging is increasing constantly, it is of critical importance to analyse those data within an accepted framework to ensure proper integration and comparison of the information collected. This book describes the ideas and procedures that underlie the analysis of signals produced by the brain. The aim is to understand how the brain works, in terms of its functional architecture and dynamics. This book provides the background and methodology for the analysis of all types of brain imaging data, from functional magnetic resonance imaging to magnetoencephalography. Critically, Statistical Parametric Mapping provides a widely accepted conceptual framework which allows treatment of all these different modalities. This rests on an understanding of the brain's functional anatomy and the way that measured signals are caused experimentally. The book takes the reader from the basic concepts underlying the analysis of neuroimaging data to cutting edge approaches that would be difficult to find in any other source. Critically, the material is presented in an incremental way so that the reader can understand the precedents for each new development. This book will be particularly useful to neuroscientists engaged in any form of brain mapping; who have to contend with the real-world problems of data analysis and understanding the techniques they are using. It is primarily a scientific treatment and a didactic introduction to the analysis of brain imaging data. It can be used as both a textbook for students and scientists starting to use the techniques, as well as a reference for practicing neuroscientists. The book also serves as a companion to the software packages that have been developed for brain imaging data analysis.

* An essential reference and companion for users of the SPM software
* Provides a complete description of the concepts and procedures entailed by the analysis of brain images
* Offers full didactic treatment of the basic mathematics behind the analysis of brain imaging data
* Stands as a compendium of all the advances in neuroimaging data analysis over the past decade
* Adopts an easy to understand and incremental approach that takes the reader from basic statistics to state of the art approaches such as Variational Bayes
* Structured treatment of data analysis issues that links different modalities and models
* Includes a series of appendices and tutorial-style chapters that makes even the most sophisticated approaches accessible ... Read more

Customer Reviews (1)

Great but who reads it?
If you do functional imaging of any kind this is a boko you need, at least for your library, because who can read this? This is very heavy literature on the subject of SPM written by the people who have developed and worked with this indispensable tool. However at times it is really incomprehensibel for normal human beings, but beautful nonetheless. ... Read more

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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.

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.

This new Second Edition incorporates numerous changes since the publication of the original 1997 edition and includes:

Features:

a new chapter on eigenvalues that covers the min-max principle, semi-classical approximation, coherent states, Lieb-Thirring inequalities, and more

extensive additions to chapters covering Sobolev Inequalities, including the Nash and Log Sobolev inequalities

new material on Measure and Integration

many new exercises

and much more ...

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.

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. ... Read more

Customer Reviews (3)

A tremendous jumpstart into modern mathematical physics
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.

A start in analysis.
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.

An excellent course of analysis with a theme
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! ... Read more

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Through numerous illustrative examples and comments, Applied Functional Analysis, Second Edition demonstrates the rigor of logic and systematic, mathematical thinking. It presents the mathematical foundations that lead to classical results in functional analysis. More specifically, the text prepares students to learn the variational theory of partial differential equations, distributions and Sobolev spaces, and numerical analysis with an emphasis on finite element methods.

While retaining the structure of its best-selling predecessor, this second edition includes revisions of many original examples, along with new examples that often reflect the authors’ own vast research experiences and perspectives. This edition also provides many more exercises as well as a solutions manual for qualifying instructors. Each chapter begins with an extensive introduction and concludes with a summary and historical comments that frequently refer to other sources.

New to the Second Edition

• Completely revised section on lim sup and lim inf
• New discussions of connected sets, probability, Bayesian statistical inference, and the generalized (integral) Minkowski inequality
• New sections on elements of multilinear algebra and determinants, the singular value decomposition theorem, the Cauchy principal value, and Hadamard finite part integrals
• New example of a Lebesgue non-measurable set

Ideal for a two-semester course, this proven textbook teaches students how to prove theorems and prepares them for further study of more advanced mathematical topics. It helps them succeed in formulating research questions in a mathematically rigorous way.

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Now revised and updated, this brisk introduction to functional analysis is intended for advanced undergraduate students, typically final year, who have had some background in real analysis. The author's aim is not just to cover the standard material in a standard way, but to present results of application in contemporary mathematics and to show the relevance of functional analysis to other areas. Unusual topics covered include the geometry of finite-dimensional spaces, invariant subspaces, fixed-point theorems, and the Bishop-Phelps theorem. An outstanding feature is the large number of exercises, some straightforward, some challenging, none uninteresting. ... Read more

Customer Reviews (2)

Five stars for content; two stars for printing
I'm a math professor in the middle of developing an introductory functional analysis course whose prerequisites will be elementary linear algebra and metric space theory and whose intended focus will be on the *abstract* theory of normed spaces and bounded linear operators between them.This book by Bollobas is about the best fit for the class that I've been able to find after an extensive search.The author covers about the right topics in about right the depth, and from what I've been able to tell it is well-organized and well-written.

During the development of this curriculum, I've been using a 1999 printing of this book borrowed from the library, and it impressed me enough that I ordered a copy of my own from Amazon, which arrived just this afternoon.To my dismay, I learned that this book, like so many of Cambridge University Press's older works, has fallen victim to the "digital printing" scourge.Contrary to what Amazon's "Search Inside!" feature indicates, the copyright page now reads "Transferred to digital printing 2004".Not that you need the copyright page to tell you that, because the loss of crispness in the typography makes that abundantly clear.And, of course, this new copy has a "perfect binding" (a misnomer if ever there was one).If you like a book with a nice professional look to it, don't get your hopes up with this one.What a shame.

Good writing, OK coverage
This undergrad text is very clear and Bollobas has a friendly style of writing. The coverage, though, is rather narrow and there are no applications. So I like Kreyszig better. ... Read more

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This second edition includes exercises at the end of each chapter, revised bibliographies, References and an Index. ... Read more