Applied Functional Analysis: Applications to Mathematical Physics, by Eberhard Zeidler, (Applied Mathematical Sciences, Vol. 108). Springer-Verlag, 1995, 479pp, DM 118.00. ISBN 0-387-94442-7.
It can be argued convincingly that of all the major areas into which we roughly subdivide modern mathematics, Functional Analysis plays the dominant role by far - at least, as long as the recent impact of a theory on the life of humankind is chosen as a dispassionate criterion.
Even the purely superficial credentials of the discipline are nothing short of striking. Serving as the foundation for quantum physics is already enough to make a theory awesome. If one adds mathematical economics and optimisation methods (the areas of application which a prominent Russian functional analyst L. Kantorovich, a Nobel Prize winner in economics himself, staunchly considered to be no less important, promising, and rich in content than theoretical physics), no other area of mathematics - be it geometry, algebra, classical analysis, or topology - can boast of such an importance. The only possible exception, and at a certain stretch besides, could be a diverse assortment of mathematical tools known as `discrete mathematics,' serving as the framework for the concept of a Turing machine, theory of computation, and thence computer science. However, there are indications that computers might themselves `go quantum' within the next couple of decades as the chips continue to be downsized to the point where quantum effects become overwhelming. Within this possible scenario, functional analysis firmly and irreversibly displaces discrete maths as the major patron of computer science as well. Che sarà, sarà.
Functional analysis could have easily reclaimed some of the most famous results in the history of mathematics, such as the independence of the continuum hypothesis. Entire branches of mathematics could have been swallowed by functional analysis: to give just one example, the entire theory of locally compact spaces (a good half of topology!) is a mere dual version of the theory of commutative C*-algebras. Some of the newest developments in mathematics are being built up on a functional-analytic foundation (Connes' noncommutative geometry and, to a large extent, quantum group theory).
In view of all this, the importance of passing a good working knowledge of functional analysis down to younger generations of researchers is great and perhaps still growing. Books like the one under review are bound to be written at regular intervals simply because of a natural market demand.
The book is published in conjunction with an (often referred to) companion volume, entitled Applied Functional Analysis: Main Principles and Their Applications. For reasons which will become clear, I would be very curious to get acquainted with this accompanying book as well.
The book under review is an ambitious project: no other description fits the goal of explaining in less than 500 pages such a variety of topics as Banach spaces and fixed point theorems with applications to integral equations and ODE's, spectral theory, approximation theory, theory of Hilbert spaces, distributions, variational problems, Fourier transform, Hilbert-Schmidt operators, self-adjoint operators, a wide collection of boundary value problems, C*-algebras and quantum statistics, scattering theory, Feynman path integral, Dirac calculus, and even some solitons and inverse scattering theory!
Unfortunately, being such an ambitious project poses the major problem for the monograph. The author tried very hard to cover as much material as possible, and the presentation is succinct to an extreme. For example, all the basic theory of C*-algebras (definition, two major sources of examples, morphisms, states, pure states, von Neumann algebras, Gel'fand-Naimark theorem) is crammed into a mere three pages: 357-359. (For comparison, it takes Dixmier at least 60 pages, to do the same in a proper way in his classic book C*-Algebras.) In such cases, every single word, every symbol, every small piece of notation suddenly become of importance, as they carry huge amounts of information and therefore are under extra strain and must be checked and double-checked as a matter of course; one expects the presentation to be much more careful and better organized than in a traditional monograph taking the reader's (and writer's) time in explaining things and able to afford an occasional slip or two. Unfortunately, this does not appear to be the case with Zeidler's book.
One of the most conspicuous inconsistencies is this: while every Banach algebra is assumed to be unital (cf. a definition on pp. 76--77), and a C*-algebra is defined as a Banach algebra with involution satisfying additional properties (p. 357), this ipso facto convention of unitality of all C*-algebras clashes with the definition of a sub-C*-algebra (p. 359), which is not required to contain the unit at all! It is well known to C*-algebra theorists how important it is to state clearly from the very beginning whether you are working in a unital or in a non-unital framework - and any reader of Zeidler's book will find themselves grossly confused.
Furthermore, the book gives the appearance of having been produced in great haste. Proposition 7 on p. 87 claims that every separable normed space can be represented as the union of an increasing sequence of finite-dimensional subspaces. One raises eyebrows, studies the proof, and eventually discovers that what the author really meant but failed to communicate was that the union of the chain of finite-dimensional subspaces is everywhere dense in the entire space, as it ought to be. OK, this was merely a lapsus linguae, one mutters to oneself, even if it should not be there at any cost, given the circumstances...
But here comes something much worse, a coup de grâce of sorts. What about the claim that the set of all continuous functions f such that f(a) = 1 is everywhere dense in C[a,b] equipped with the supremum norm (exercise 1.1.d, p. 92), and that the same is true for the set of functions f with f(a) > 0 (1.1.c)? The former statement is especially mind-boggling, because by combining it with an earlier exercise 5 on p. 30, the poor reader comes to an immediate conclusion that every continuous function on a closed interval [a,b] must vanish at the left endpoint! Ironically, the author himself contentedly refers to his choice of exercise problems as "a carefully selected collection" (Introduction, p. xvi).
I could go on and on for quite a while. The only thing I can make out of it all is that, sadly, the publishers seem not to have ensured adequate scrutiny of the manuscript of the book before accepting it for publication
Having now earned a copy of Zeidler's book through reviewing it for our Society, I feel with some regret that the new acquisition for my very modest office book collection is of a somewhat doubtful value. (Would you use such a book for learning new things after it proved to have such flaws in presenting things that you already knew? Would you recommend it to your students? Would you cite it in your research papers as a reliable reference source?) Certainly, I will stick as ever before to Reid and Simon's four volumes of Methods of Modern Mathematical Physics written with so much care and effort invested in their work and containing complete proofs of statements of results, which are, incidentally, all correct.
At the very beginning of his book, Eberhard Zeidler claims that "there are two ways of teaching mathematics, namely, (i) the systematic way, and (ii) the application-oriented way," and stresses that "the present book is based on the second approach." However, the reviewer was always sceptical about categorizing the mathematics that we produce and teach to our students otherwise than good and poor. In particular, the poor remains such even if labelled applied. The book under review only substantiates this conviction.
Vladimir Pestov, Victoria University of Wellington
Algebraic Topology. A First Course, by William Fulton (Graduate Texts in Mathematics Vol. 153). Springer-Verlag, 1995, 435pp, DM98.00. ISBN 3-540-94326-9.
William Fulton is particularly known as a prolific author of good mathematical books, and the introductory text on topology under review fully meets one's expectations. If the present reviewer were ever to teach an Honours course in topology in New Zealand, he would have probably selected the text under review as a textbook. It was written on the basis of an undergraduate course taught by the author at Brown University and the University of Chicago, which means that it would perfectly suit the needs of an Honours course at Victoria; the lecturer would have to spend some time providing a necessary background in basic analysis which is not part of the undergraduate curriculum, such as theory of differential forms (the task is greatly facilitated by the presence of a number of appendices containing the fundamentals from analysis and algebra needed in the course), but one's efforts would pay off. Browsing the book is ever delightful, and the targeted audience of it -- "students of mathematics or science who are not aiming to become practicing algebraic topologists" -- means that the presentation is both extremely intelligible and careful.
The contents of Fulton's book bring back to life the sweetest memories of this reviewer's PhD student years spent in the Department of Geometry and Topology at Moscow State University. Differential forms, vector fields, flows in Euclidean spaces; deformations and
homotopies; winding numbers, with applications to the fundamental theorem of algebra and Borsuk's fixed point theorem; degrees of maps and the fundamental group of the circle; cohomology, homology, their computation for basic examples, De Rham's theorem; deformation retracts; indices and singularities of vector fields; vector fields on spheres and other surfaces; Poincaré-Hopf theorem; the Euler characteristic; Cauchy Integral Theorem; residue theorem; the Mayer-Vietoris exact sequences; covering spaces, fundamental groups, homotopy lifting, deck transformations; fundamental group and first homology group; universal covering; Van Kampen theorem; orientation; triangulation and classification of compact oriented surfaces; the fundamental group of a surface; Riemann surfaces and branched coverings; Riemann surfaces and algebraic curves; the Riemann-Roch and Abel-Jacobi theorems; higher dimensions; higher homology; duality between homology and cohomology; finally, an outline of simplicial complexes.
A large number of exercises and pictures represent especially attractive additional features of the book.
I like the book and highly recommend it to all teachers of beginners' courses in algebraic topology, as well as to those teaching themselves the basics of the subject (which is, after all, an essential part of the general mathematical culture).
Vladimir Pestov
Victoria University of Wellington
Modern Analysis and Topology, by Norman R. Howes. Universitext, Springer-Verlag, New York, 1995, 403pp, DM 68.00. ISBN 0-387-97986-7.
This book is designed to provide an integrated development of modern analysis and topology through the integrating vehicle of uniform spaces. It is intended that the material be accessible to a reader of modest background. An advanced calculus course and an introductory topology course should be adequate. But it is also intended that this book be able to take the reader from that state to the frontiers of modern analysis and topology in-so-far as that can be done within the framework of uniform spaces. This is an experiment worth the effort, but opinions will differ on how successful the author has been.
Modern analysis is usually developed in the setting of metric spaces although a great deal of harmonic analysis is done on topological groups and much of functional analysis is done on various topological structures. All of these spaces are special cases of uniform spaces.
On the other hand, modern topology often involves spaces that are more general than uniform spaces. It is the view of the author that "uniform spaces provide a setting general enough to investigate many of the most important ideas in modern topology, including the theories of Stone-Cech compactification, Hewitt Real-compactification and Tamano-Morita Para-compactification, together with the theory of rings of continuous functions, while at the same time retaining a structure rich enough to support modern analysis".
The best way of indicating the range of topics considered in this book is probably by listing the chapter headings. The first seven chapters are topological in essence while the last five are mostly analysis.
1. Metric spaces
2. Uniformities
3. Transfinite sequences
4. Completeness, cofinal completeness and uniform paracompactness
5. Fundamental constructions
6. Paracompactifications
7. Real-compactifications
8. Measure and integration
9. Haar measure in uniform spaces
10. Uniform measures
11. Spaces of functions
12. Uniform differentiation
It seems that much, perhaps even a half, of the material in this book has not appeared before in book form. So there is plenty to interest the expert as well as the novice.
An attempt has been made to document the history of all the central ideas, so references and historical notes are embedded in the text. These can lead the interested reader to the foundational sources where these ideas emerged. But the lack of a listing of all references in a bibliography is a major shortcoming. It infuriated this reviewer.
Ivan Reilly, University of Auckland
SPRINGER-VERLAG PUBLICATIONS
Information has been received about the following publications. Anyone interested in reviewing any of these books should contact
David Alcorn
Department of Mathematics
University of Auckland
(email: alcorn@math.auckland.ac.nz)
Applied Mathematical Sciences
108. Zeidler E Applied functional analysis. Main principles and their applications. 440pp.
Encyclopaedia of Mathematical Sciences
12. Novikov SP (ed) Topology I. General survey. 319pp.
35. Shafarevich IR (ed) Algebraic geometry II. Cohomology of algebraic varieties. Algebraic surfaces. 264pp.
49. Parshin (ed) Number theory. I. Fundamental problems. Ideas and theories. 303pp.
50. Arhangel'skii AV (ed) General topology II. Compactness. Homologies of general spaces. 256pp.
51. Arhangel'skii AV (ed) General topology III. Paracompactness. Function spaces. 229pp.
57. Kostikin AI (ed) Algebra VI. Combinatorial and asymptotic methods in algebra. Nonassociative structures. 287pp.
65. Shubin MA (ed) Partial differential equations VIII. Overdetermined systems. Dissipative singular Schrödinger operator. Index theory. 258pp.
77. Kostrikin AI (ed) Algebra IX. Finite groups of Lie type. Finite-dimensional division algebras. 239pp.
Graduate Texts in Mathematics
80. Robinson DJS A course in the theory of groups. (2nd ed). 505pp.
161. Borwein P Polynomials and polynomial inequalities. 465pp.
162. Alperin JL Groups and representations. 194pp.
Undergraduate Texts in Mathematics
Anglin WS The heritage of Thales. 282pp.
Axler S Linear algebra done right. 250pp.
Browder A Mathematical analysis. 345pp.
Childs LN A concrete introduction to higher algebra. (2nd ed) 500pp.
Elaydi SN Introduction to difference equations. 380pp.
Exner G An accompaniment to higher mathematics. 225pp.
Hairer E Analysis by its history. 373pp.
Miscellaneous
Baumann G Mathematicareg. in theoretical physics. 348pp
Diener F (ed) Nonstandard analysis in practice. 250pp.
Dodson CTJ Experiments in mathematics using Maple. 465pp.
Gerber HU Life insurance mathematics. (2nd ed) 217pp.
Ribenboim P The new book of prime number records. (3rd ed) 500pp.
Software Review
Ray Hoare from Hoare Research Software has been marketing a number of mathematical programs, such as Mathcad, Mathematica, Maple and Matlab. However, much of the use of these programs is by people in science and engineering who use mathematics as a tool, rather than by mathematicians. Relatively few people are using them for teaching or developing mathematics.
A new program has just been released by MathSoft, called StudyWorks. It is explicitly designed for teaching maths and physics. Have a look at http://www.mathsoft.com/studyworks/prodinfo/proddes.htm to see the manufacturer's hype about it.
Ray is not a mathematics teacher, so can't really assess whether it meets a need for the NZ curriculum. He would like to hear from people who can comment on the hype, or would like to evaluate the full program. A free copy is available for the purpose of doing a review article suitable for publication in an appropriate NZ journal. Contact Ray at ray@hrs.co.nz, or phone 07 839 9102.
Hoare Research Software Phone +64 7 839 9102
P.O. Box 4153 Fax +64 7 839 9103
Hamilton East, New Zealand Email info@hrs.co.nz
WWW page - http://www.hrs.co.nz/comm/hrs