## Reviews of John L Kelley's books

We give below extracts from some reviews of John L Kelley's books. These books cover a wide range from high school texts to research monographs.

**1. Exterior ballistics (1953), by E J McShane, J L Kelley and F V Reno.**

**1.1. From the Preface.**

As a rule the authors of a scientific book presumably hope that it will prove useful. The authors of this book wish devoutly that it will turn out to be quite useless, and that the application of exterior ballistics, with lethal intent, may cease. Nevertheless it is clear that armies will mass and nations stand in readiness for war until "Homo sapiens" succeeds in better deserving his self-bestowed name. While this endures, there must be many who know something about the flight of projectiles, and a few who know much about it. For these, this book is written. To the optimist who feels that this book is pointless because there will never be another war we can say only that we hope he is right. To the pessimist who feels that this book is pointless because the next war will be fought with weapons of such wide destructive power that it matters little where they are delivered, we would say that few weapons indeed, having once been useful, have been entirely discarded. The battle-axe survived in the hatchet of the commandos; the spear survived in the bayonet and even sticks and stones killed many in the first days of the independence of India. If there is another war, we may feel reasonably sure that guns, bombs and rockets will not be useless.

**1.2. Review by: Leslie Howarth.**

*Bull. Amer. Math. Soc.*

**60**(3) (1954), 274-276.

This book forms a welcome addition to the very limited number of works on exterior ballistics; had there been many more competitors I suspect its welcome would have been equally warm. The first chapter, roughly one-sixth of the book, is devoted to mathematical and physical preliminaries and is intended "to make the book intelligible to anyone who has had a reasonably good undergraduate course either in mathematics or physics." ... Coming now to exterior ballistics proper the treatment starts in Chapter II by a discussion of the aerodynamic forces acting on the projectile ... Chapter III is devoted to methods of determining drag from velocity and retardation measurements, to wind tunnel methods of determining drag, lift, and overturning moment coefficients and to damping experiments in the wind tunnel for providing the yawing moment due to yawing which by variation of the axis of oscillation can be made to give the cross force due to cross spin. Chapters IV to IX are devoted to the "normal" equations of the trajectory, their solution and their differential effects. ... Apart from a short chapter on bomb-sights and an historical appendix, the rest of the book is effectively devoted to the angular motion of projectiles spun and unspun, and its effect on the C.G. motion; it is here that the book will, I think, capture most attention. I found it a most stimulating account of the subject and would single out for special mention the treatment of stability.

**1.3. Review by: Robert A Rankin.**

*Mathematical Reviews*MR0060355

**(15,657h).**

This long awaited volume is destined to become a classic and to take a worthy place alongside earlier notable works on ballistics. In it we have a text-book on exterior ballistics in which the subject is developed logically and rigorously from first principles, where assumptions when justifiable are justified and when not justifiable are fairly discussed, instead of the vague appeals to 'well-known' principles of mechanics, the incomplete analogies, the surreptitious dropping of awkward 'negligible' terms and the failure to separate the mathematical from the empirical which characterise many earlier books, and which are the despair of the critical reader. Because of this the book is not made harder to read but easier. Even the reader with relatively meagre mathematical equipment should be capable of understanding nearly everything if he is willing to apply himself and makes good use of the excellent first chapter, which is a concise compendium of all the techniques which are used in the sequel.

**2. General Topology (1955), by John L Kelley.**

**2.1. Review by: Peter J Hilton.**

*The Mathematical Gazette*

**41**(336) (1957), 156-157.

"This book is a systematic exposition of the part of general topology which has proven useful in several branches of mathematics. It is especially intended as background for modern analysis, and I have, with difficulty, been prevented by my friends from labelling it: What Every Young Analyst Should Know." The friends were surely right; for while the book is clearly of the greatest value as a work of reference, the main question in the reviewer's mind was to decide to what reader it should be recommended. ... The author admits the difficulty of combining in one book the function of reference and text; but, though the book is based on "various lectures" by the author and is logically self-contained, the abstractness of the formulations and the generality of the statements of many of the theorems (the author is at pains to avoid point separation axioms) would certainly trouble an audience of British mathematicians quite inexperienced in topology. It would seem then that the book is suitable for the analyst or for the student of topology who has had some grounding in such a work as Kuratowski's

*Topologie*(Vol. I) or Newman's

*Topology of Plane Sets*. Indeed, for such a reader, the book can be highly recommended for its clarity and the vividness of the presentation. These are enhanced by the author's style which, though it might be described as "racy", appealed strongly to the reviewer. ... the last word should be that this book made most enjoyable reading.

**2.2. Review by: Edwin Hewitt.**

*Bull. Amer. Math. Soc.*

**62**(1) (1956), 65-68.

The appearance of a comprehensive treatise in English on present day set-theoretic topology is an important event. The rapid progress in set-theoretic topology during the past 20 years, and the ever increasing applications of this discipline to analysis, make the appearance of the volume under review particularly appropriate at the present time. Professor Kelley has set himself the task of producing a book useful for both students and specialists, and he has succeeded to a remarkable extent in reaching both of these somewhat inconsistent goals. Like many other texts of this genre, the present volume could be understood, at least in theory, by any intelligent person who can read English. All of the machinery is supplied; but a knowledge of the real numbers and elementary abstract algebra as set forth for example in Birkhoff-MacLane (

*A survey of modern algebra*, 1953) and a thorough knowledge of elementary analysis are certainly minimal prerequisites for appreciating this book. The author's style is spirited, to say the least. The atmosphere of an informal and humorous lecture pervades the book, especially in the first part. An essential difference between oral and written communication is well illustrated here, for some sallies that would clearly enliven a lecture are less felicitous in print.

**2.3. Review by: Francis P Larkin.**

*The Journal of Symbolic Logic*

**27**(2) (1962), 235.

The bulk of the material presented in this work is, of course, outside the field of this Journal. Chapter 0 consists principally of a discussion of elementary set-theoretic matters, acquaintance with which is presupposed throughout the rest of the book. The treatment is clear, but quite concise. The Appendix furnishes a rigorous backstopping of Chapter 0 by means of a system of axiomatic set theory which is there credited to A P Morse and which does not seem to have been discussed hitherto in the literature.

**2.4. Review by: Arthur Harold Stone.**

*Mathematical Reviews*MR0070144

**(16,1136c)**.

This text-book is intended to provide a background for "modern analysis'' (which seems to mean the theory of linear spaces). ... Throughout the book, the treatment reveals modern refinements, which are shown partly in the elegance of the methods, and partly (perhaps less significantly) in the weakening of assumptions. ... a valuable feature of the book is the collection of exercises; besides routine exercises, these include further (often extensive) theoretical developments and applications (with suitable hints), and some useful counterexamples. There is very little on the geometrical aspects of the subject; local connectedness receives little more than mention, and arcs are not mentioned at all. ... The author has been more concerned to expound the main ideas and methods, and to illustrate the wide range of applications, than to treat any topic very thoroughly. His style also places emphasis on essentials at the expense of details. There are many illuminating comments, in a conversational style, which make the arguments easy to follow in the large. However, the speed with which the foundations are laid, and the not infrequent omission of small steps in the reasoning and of "obvious'' routine theorems, may sometimes puzzle a beginner.

**3. Introduction to Modern Algebra, Official Textbook for Continental Classroom (1960), by John L Kelley.**

**3.1. From the Preface.**

The text is designed for a first-year course in a liberal arts college program, or for a course for high school students in an accelerated program. Many recommendations of professional groups concerned with the mathematical curriculum have been embodied in the text, and it may, therefore, be useful for teachers of high school mathematics. Some of the problem lists are quite long, and many of the problems are substantial mathematical theorems. The material which is covered in the text comprised a completely unorthodox course in college algebra.

**3.2. Review by: Bruce E Meserve.**

*The Mathematics Teacher*

**54**(5) (1961), 370-371.

This book is concerned with axioms for the real numbers, elementary concepts of sets and statements, vectors as ordered pairs and ordered triples of real numbers, complex numbers as ordered pairs of real numbers, the algebra of vector spaces, and matrices. Applications of each concept are made to a variety of problems, including many problems normally considered in high school. The point of view is that of abstract algebra; the scope of topics includes many items that high school teachers think of as geometry. In this sense, there is an emphasis upon the interrelations of algebra and geometry. Theorems and proofs are emphasized through out the book. Figures and illustrative examples are used freely. ... In its present form, the book is hard reading for teachers with modest training, but worthy of serious study for mathematics teachers with at least average training. High school students with special training or interest will find many topics interesting. The author has a fine sense of humour. The style of exposition is frequently easy going. Each reader will find many problems of special interest.

**4. Linear Topological Spaces (1963), by J L Kelley and I Nanioka.**

**4.1. From the Foreword.**

This book is a study of linear topological spaces. Explicitly, we are concerned with a linear space endowed with a topology such that scalar multiplication and addition are continuous, and we seek invariants relative to the class of all topological isomorphisms. Thus, from our point of view, it is incidental that the evaluation map of a normed linear space into its second adjoint space is an isometry; it is pertinent that this map is relatively open. We study the geometry of a linear topological space for its own sake, and not as an incidental to the study of mathematical objects which are endowed with a more elaborate structure. This is not because the relation of this theory to other notions is of no importance. On the contrary, any discipline worthy of study must illuminate neighboring areas, and motivation for the study of a new concept may, in great part, lie in the clarification and simplification of more familiar notions. As it turns out, the theory of linear topological spaces provides a remarkable economy in discussion of many classical mathematical problems, so that this theory may properly be considered to be both a synthesis and an extension of older ideas.

**4.2. Review by: Fred E J Linton.**

*Amer. Math. Monthly*

**72**(2) (1965), 218-219.

The theory of real or complex topological vector spaces is here developed with minimal algebraic and topological assumptions as well on t he structure of the space as on the preparedness of the reader. The first chapter, ignoring topological questions, presents the basic order and convexity results that essentially distinguish real or complex vector spaces from arbitrary abelian groups. The second chapter, after a review of uniform topology, developed in the special context of vector spaces, discusses the interplay between convexity and boundedness. Chapter Three displays the basic theorems dependent on Baire category arguments. The fabrication of tools ends in the fourth chapter, where such inherently topological convexity results as the separation, extension, and extreme point theorems appear. The fifth, last, and richest chapter deals with duality. ... In typical Kelley fashion, it is the exercises that float the reader into the deepest water of the theory. There are whole, well-organized sequences of problems developing, in part, Hilbert space, distributions, closed graph theorems, Banach-Steinhaus theorems, direct and inverse limits, tensor products, spaces of analytic functions, spaces of power-integrable functions, completely continuous operators, Fréchet, Mantel, and Köthe spaces, and of course dozens of examples and counterexamples. Graduate students could easily do worse than to spend their time working out these exercises and studying the clearly exposed text. And, as was the case with Kelley's earlier

*Topology*, the present book will probably find a place on every working analyst's shelf simply because of the many definitive statements of (often inaccessible) fact densely scattered throughout the text and problems.

**4.3. Review by: George T Roberts.**

*The Mathematical Gazette*

**50**(371) (1966), 75-76.

This book is about linear spaces which are endowed with a topology, which need not be that obtained from a norm. This theory was developed in the 1940s and early 1950s notably by G W Mackey and various members of Bourbaki. This text, of which a first draft was prepared towards the end of that period, is a systematic account of that work. It was revised since then, but contains little about any more recent discoveries in this field. ... An assessment of the value of a book such as this depends on the purpose it is intended to serve. As a textbook for a student approaching the subject for the first time, it suffers from the defect that it has long passages of preliminary material during which it would be impossible for him to discover where it was all leading to. This is to some extent mitigated by orienting paragraphs before each section. (Indeed many sections start with a summary in small type followed by another summary of about twice the length in normal type: a practice the reviewer found confusing.) On occasions the simple is made to seem difficult. For instance the definition of weak topology would make it seem quite an intricate concept. However, since one of the co-authors is an author of a book on the same topic obviously more adapted to the needs of beginners, it is reasonable to assume that this text is not aimed principally at this readership. A good reference book in this line has been needed for some time. This book will go a long way towards filling this gap.

**4.4. Review by: Richard Arens.**

*Mathematical Reviews*MR0166578

**(29 #3851)**.

This book is the result of the efforts of many authors who happened to agree (in 1953) on how to arrange the theory of topological linear spaces in such a way as to make the more recent (at that time) results on duality appear as the natural consequences of the preliminary work. ... It is remarkable that such a joint effort should have produced such a valuable book, which will probably be the best text for a graduate course covering all that material, excepting operators in Hilbert space and topological linear algebras, which an aspirant to functional analysis ought to learn. This is in a large part due to the excellent and non-routine exercises, mainly added when the manuscript was revised in 1961.

**5. Algebra: A Modern Introduction (1967), by J L Kelley.**

**5.1. Review by: Roger F Wheeler.**

*The Mathematical Gazette*

**55**(391) (1971), 110-111.

This book is a revised and expanded version of an earlier work Introduction to Modern Algebra by the same author in the same series (the University Series in Undergraduate Mathematics). It is "intended to provide a text for a standard pre-calculus course in algebra" and so in Britain, at any rate, there is no large group of students who could be expected to have a similar background to those Professor Kelley had in mind. (There are, for example, a few passing references in the book to trigonometry, made almost apologetically, with the assumption that the reader will be daunted by the appearance of sin and cos.) The great merit of the book is that it is genuinely written for students and not for teachers. The student is apostrophized and the style in places is quite racy. The mathematical pace, however, is leisurely and the author is continually striving to reassure and encourage his readers. At times, indeed, this aim is carried to excess, with pseudo-ingenuous remarks such as "I'm not quite sure what a parameter is, but I think that you call a number a parameter when you're thinking about a function whose domain is a set of numbers." One certainly commends the author's attempt to make direct contact with the student reader, but one must admit that, in this case, the result is not entirely satisfactory. In the welter of explanation and apology, essential points seem to get lost or glossed over. Some students would certainly be irritated by the slow pace and the author's assumption that the reader is a somewhat reluctant mathematician. On the other hand, some students would certainly profit by reading this fairly concrete treatment before embarking on more abstract books, which might make less concessions to their frailty.

**6. Elementary Mathematics for Teachers (1970), by John L Kelley and Donald Richert.**

**6.1. Review by: Max S Bell.**

*Amer. Math. Monthly*

**79**(1) (1972), 102-103.

This book for teachers and prospective teachers of the first six school grades consists mainly of a treatment of the real number system and its subsystems, with a couple of chapters at the end giving a quick romp through functions, rudiments of school algebra, and measure of length, area, volume, and weight. If you are teaching in one of those apparently rare places offering the full CUPM recommended mathematics sequence for elementary school teachers you would do well to examine this as a prospective text for the early part of the sequence. You should be warned, however, that it is an exasperating book, with a number of excellent features that would warrant its use but also with a number of flaws that would argue against its use. ... on balance I found this a most interesting book and I recommend that teachers of courses for elementary school teachers get it and read it for its virtues. Whether its flaws disqualify it for classroom use each must decide for himself.

**7. Measure and Integral: Volume 1 (1988), by John L Kelley and T P Srinivasan.**

**7.1. From the Preface.**

This is a systematic exposition of the basic part of the theory of measure and integration. The book is intended to be a usable text for students with no previous knowledge of measure theory or Lebesgue integration, but it is also intended to include the results most commonly used in functional analysis. Our two intentions are some what conflicting, and we have attempted a resolution as follows. The main body of the text requires only a first course in analysis as background. It is a study of abstract measures and integrals, and comprises a reasonably complete account of Borel measures and integration for R. Each chapter is generally followed by one or more supplements. These, comprising over a third of the book, require some what more mathematical background and maturity than the body of the text (in particular, some knowledge of general topology is assumed) and the presentation is a little more brisk and informal. The material presented includes the theory of Borel measures and integration for R

^{n,}the general theory of integration for locally compact Hausdorff spaces, and the first dozen results about invariant measures for groups. Most of the results expounded here are conventional in general character, if not in detail, but the methods are less so.

**7.2. Review by: Pedro Jiménez Guerra.**

*Mathematical Reviews*MR0918770

**(89e:28001)**.

In this book the authors give a systematic exposition of the basic part of the theory of measure and integration. Most of the results are conventional in their general character, but the methods are less so.

**7.3. Review by: David Applebaum.**

*The Mathematical Gazette*

**73**(465) (1989), 271-272.

This is the first part of a two-volume work which is advertised as being a suitable basis for an undergraduate course for students possessing a background in elementary real analysis. This book covers the basic theory and is organised so that each chapter concludes with one or more supplements describing additional topics which lie outside the core material. The subsequent volume will contain all the exercises together with additional supplements. Modern theories of integration all derive from Lebesgue's approach which is superior to the older idea of Riemann integration in that a larger class of functions are integrable and one so obtains the semi-normed

*L*

^{p}spaces which have the virtue of being complete. Indeed the Lebesgue integral has become an essential tool in many areas of mathematics such as functional analysis, probability theory, rigorous quantum theory and harmonic analysis. The usual approach to Lebesgue integration is to define the integral of the indicator function of a measurable set to be its measure and then extend to more general functions by linearity and a limiting procedure. Traditionally one first studies the Lebesgue measure on the real line and the associated integral before moving on to the more general formalism of measurable spaces and

*σ*-algebras. The book under review takes a somewhat more sophisticated and abstract approach. ... A major weakness of this book as an introduction to the subject is the lack of examples. ... On the other hand, this would be an excellent book on which to base a course for graduate students in analysis (or a related area) who have already studied a basic course on measure theory and require further instruction at a higher level. Indeed, many of the supplements which cover such topics as Haar measure on locally compact groups and the Bochner integral for Banach space valued functions would be of obvious value in this context.