## Reviews of Tom Flett's mathematics books

Thomas Flett wrote two mathematics books, the second of which was published after his death. We give below short extracts from some reviews of these books and extracts from the Preface.

**Mathematical Analysis (1966), by T M Flett.**

**1.1. From the Preface:**

This book provides an introduction to the theory of functions of one and several real variables, covering the material which is usually presented in the first two years of an undergraduate course. I have tried to present the material so that it appears naturally in the context of modern mathematics, and to indicate the directions of further developments. I hope that the book will thus serve as an introduction to both modern and classical analysis. I have assumed that any student beginning this book will be familiar with the techniques of elementary calculus, and that while reading the book he will be following a parallel course on abstract algebra. However, a knowledge of abstract algebra is not essential until Chapter 10, where extensive use is made of elementary linear algebra, particularly concerning the Euclidean spaces. The material needed there is summarized in §6 10.2 and §6 10.10. Chapter 1, which should be read "with judicious lightness", is essentially a dictionary to establish both our starting point and our usage concerning sets, functions and numbers. I have included here certain topics, such as the natural numbers, finite and infinite sets, and inductive definition, which are often taken for granted. It has been my experience that many students find difficulty in the study of analysis because they are uncertain of this basic material. I have used that definition of function (as a set of ordered pairs) which seems most appropriate to elementary analysis, and I have tried throughout the book to keep the clear distinction between a function and its values which is necessary in more advanced work involving sets of functions.

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

*The Mathematical Gazette***51**(375) (1967), 88.

This is a careful, detailed and complete treatment of the theory of functions of one and several real variables covering the material usually presented in the first two years of an undergraduate honours course. Although the number of pages is less than in the book by Scott and Tims reviewed below, the treatment is less discursive, and considerably more maturity is demanded from the student. The first chapter is a closely packed account of fundamental notions such as those of function, inverse, restriction, composition, inductive definitions, sequences, and countable sets; one somewhat surprising omission is any mention of logical connectives or quantifiers. Starting from the assumption that a set of real numbers bounded above has a least upper bound, the author discusses continuity and limits for real and complex functions of a single real variable. In the definitions of these concepts the idea of a neighbourhood is used in place of the more traditional definitions in terms of inequalities. ... The penultimate chapter contains a very full treatment of all the essential properties of topological spaces, such as Hausdorff spaces, metric spaces, connectedness, compactness, uniform properties and completeness. The reviewer found the last chapter, which deals with vector-valued functions of several real variables, particularly interesting and impressive. In it the author deals in a modern manner with the usual topics, such as differentials, Jacobians, the chain rule and the implicit function theorem. ... This is definitely one of the best books on the subject to have appeared in recent years. The subject matter is not easy, the style is concise, but the exposition is lucid.

**1.3. Review by: Tom M Apostol.**

*Mathematical Reviews*, MR0210836**(35 #1721)**.

This carefully written book is intended as a course in analysis for students familiar with elementary calculus techniques. The first chapter discusses fundamental concepts such as sets, functions, natural numbers, induction, and countability. The second treats real and complex numbers. This is followed by six chapters dealing with calculus of real-and complex-valued functions of one real variable (continuity, limits, derivatives, Riemann integrals, infinite series, and uniform convergence). Up to this stage most of the topics treated are from elementary calculus. ... In the last two chapters the subject matter jumps suddenly to a considerably more advanced level. Chapter 9, entitled "Topological spaces", deals with neighborhood systems, Hausdorff spaces, metric spaces, connectedness, and compactness. The tenth and last chapter, which pre-supposes some knowledge of linear algebra, discusses differential calculus of vector-valued functions of several variables. It includes a thorough and novel treatment of differentials, Jacobians, the inverse and implicit function theorems, constrained maxima and minima, and functional dependence.

**Differential analysis (1980), by T M Flett.**

**2.1. Preface by John S Pym.**

On 13 February 1976 Professor T M Flett died at the early age of 52. At that time he had almost completed the manuscript of the present book. In order that so much effort should not be lost, I undertook the task of finishing the work. My guiding principle has been that the book is still Professor Flett's: although I have no doubt that he would have made alterations in arriving at his own final version, I am sure that the reader would wish to hear his voice, even imperfectly, rather than mine. The text he left has been altered only where there were clear indications that he himself had intended to do this or on the rare occasions when errors had crept in. A few parts of the book which he clearly proposed to include did not exist even in manuscript. The reader will wish to know that I am solely responsible for §6 2.13, for the historical note on differentials (§6 4.7) and for the notes on chapters 3 and 4.

**2.2. Review by: Walter Kurt Hayman.**

*The Mathematical Gazette***65**(433) (1981), 232-233.

This book was left in an almost complete form by the author on his early death in 1976, and completed by Professor Pym in what was clearly a labour of love. It deals with the general differential properties of functions from one vector space into another. There are four chapters. In the first chapter on functions of one real variable the classical theorems of Rolle, Taylor etc. are developed. However the development is sophisticated. ... The next chapter deals with existence and uniqueness for an ordinary differential equation ... The author discusses in a most scholarly way the historical developments, and there is an enormous number of references from the 18th century to the present day. This is in some ways the best part of the book since an outsider occasionally feels that the value of the results obtained does not match the complexity of the machinery. In view of the specialised nature of the material the price is reasonable.

**2.3. Review by: Stuart J Sidney.**

*American Scientist***68**(4) (1980), 460-461.

Here is a scholarly treatment of the differential analysis of functions with values (and, most often, with independent variables as well) in normed spaces. Both the theory and the applications (to differential equations, calculus of variations, etc.) are developed in great depth, many nontrivial exercises are very well chosen to amplify the material further, several historical notes enliven the discussion by pointing out the holes in the works of the old masters and how their successors filled them, and the bibliography and index are comprehensive enough to enlarge the book's value as a reference. While the style is somewhat dry, it is clear; discussion is careful and coverage complete. An unusual feature is the inclusion of the Hadamard differential and the proof that it yields the most general reasonable notion of differentiation for which the chain rule is valid.

**2.4. Review by: Theodore S Bolis.**

*Mathematical Reviews*, MR0561908**(82e:26021)**.

This book presents the generalization and the ensuing application of the fundamental theorems of the differential calculus of functions of one real variable to the case when the domain or the range (or both) become subsets of normed spaces. The book is published posthumously. J S Pym finished and edited the almost completed manuscript. ... Each chapter ends with extensive and very interesting historical notes. There are six appendices containing the basic definitions and results needed from functional analysis. Further historical-bibliographical notes are included before the extensive bibliography. The book is concise, elegant and of interest to mathematicians of every persuasion. General abstract results of the theory are put to immediate use in "hard" areas of mathematics such as differential equations and the calculus of variations. Thus the reader can realize that the seemingly far-fetched generalizations of the familiar theorems of the differential calculus not only satisfy the curious theoretical mathematician, but also have a utilitarian aspect.