## Reviews of Roger Godement's books

We give below extracts from some reviews of some of Roger Godement's books. We list these in order of publication of the first edition. When an English translation of a text originally published in French is listed we give it below the French text.

**1. Topologie Algébrique et théorie des faisceaux (1958), by Roger Godement.**

**1.1. Review by: J H C Whitehead.**

*The Mathematical Gazette*

**44**(347) (1960), 69-70.

The theory of sheaves (faisceaux) is one of the outstanding developments in mathematics during the last twenty years. ... The book under review is the first to be written on the subject. Algebraic topology and related subjects have been expanding so rapidly during the last fifteen years that any book on an advanced level has been likely to be obsolete before it was printed. The book under review has certainly escaped this fate. In fact, written in the light of "Homological algebra" (Cartan and Eilenberg) and of Grothendieck's paper, it is as timely a book as any I can remember. Moreover it is very good.

**1.2. Review by: R Deheuvels.**

*Mathematical Reviews*, MR0102797

**(21 #1583)**.

This excellent book is presented by the author as the first part of a two-volume work, which will make for its size an effective treatise on algebraic topology (considered from a different perspective from the "geometric" point of view of Eilenberg-Steenrod.) The main purpose of this first volume is to deal with the cohomology of any topological space with coefficients in a sheaf ... and also to prepare the content of the second volume (Steenrod operations, etc. ...), by a detailed study of theory of products and simplicial structures. ... The breadth of ideas, clarity and effectiveness of the presentation make it an excellent book.

**2. Variétés différentiables. Résumé de leçons (1959), by Roger Godement.**

**2.1. Review by: L A Santaló.**

*Mathematical Reviews*, MR0110063

**(22 #946)**.

This [51-page] résumé of lectures given by the author at the University of Recife (1956) is a very lucid exposition of the fundamental concepts on differentiable manifolds which contains the following topics: 1. Exterior algebra; 2. Differentiable functions; 3. Differentiable manifolds; 4. Tangent vectors and differentials; 5. Fiber spaces.

**3. Cours d'Algèbre (1963), by Roger Godement.**

**3.1. Review by: E A Maxwell.**

*The Mathematical Gazette*

**47**(362) (1963), 364-365.

I ought perhaps to begin by explaining how it is that I come to be reviewing this book. It arrived to me as Editor, and I glanced idly at it, wondering what reviewer to tackle. But my glance became less and less idle as I began to feel that here was just the book that we have been wanting, and I now recommend it without reserve as one of the most exciting texts I have met for many years. ... Here, it would seem, is everything: sets and functions, groups, rings, fields, complex numbers, vector spaces, linear mappings, determinants, polynomials, algebraic equations, matrices, hermitian forms ... [The] book is beautifully produced, the style is a joy and everyone who is looking for learning without tears in what is, admittedly, a difficult field will find here an invaluable source of inspiration.

**3.2. Review by: K Nomizu.**

*Mathematical Reviews*, MR0158884

**(28 #2106)**.

This book, based on the author's course at the University of Paris, covers the basic subjects of modern algebra which, according to the author, everybody considers indispensable for future mathematicians or physicists. They are presented in very general setting and in a lucid, rigorous style. ... In the reviewer's opinion, the author's usage of "exemples", where he mostly introduces standard and useful models of concepts or structures and never wastes time by giving arbitrary so-called numerical examples, illustrates the high level of exposition. On the other hand, exercises ranging over 160 pages include many computational problems as well as more difficult ones in which the author gives supplementary and advanced results ...

**4. Zeta functions of simple algebras (1972), by Roger Godement and Hervé Jacquet.**

**4.1. Review by: L Corwin.**

*Mathematical Reviews*,

**MR0342495**

**(49 #7241)**.

Two of the most important sorts of zeta-functions in number theory are those of Artin and Hecke. Indeed, class field theory can be described (from an analyst's point of view, at least) as the attempt to show that they are the same. The Artin zeta-functions (or L-series) are easy to define; those of Hecke are not. The aim of the authors is to define the Hecke zeta-functions for all simple algebras over algebraic number fields and to prove a functional equation for them.

**5. Introduction à la théorie des groupes de Lie (2 volumes) (1982), by Roger Godement.**

**5.1. Review by: G A Reid.**

*Mathematical Reviews*,

**MR0679859**

**(85i:22001a)**, MR0679859

**(85i:22001b)**.

This book, in two volumes, is based on a course of lectures given by the author at the University of Paris in 1973-74 and provides a comprehensive introduction to the theory of Lie groups. Starting from a knowledge of the fundamentals of linear algebra and general topology, the reader is carried along (at a brisk pace) through other necessary basic material, including that relating to differentiable manifolds. ... The scheme of the book is to deal first with linear groups, examining their analytic structure through that of the general linear group. It then proceeds to the general theory of Lie groups, defined as topological groups having the structure of a differentiable manifold such that the group operations are differentiable maps.

**6. Analyse Mathématique I. Convergence, fonctions élémentaires (1998), by Roger Godement.**

**6.1. Review by: Gerald B Folland.**

*Amer. Math. Monthly*

**114**(2) (2007), 172-176.

Let's indulge in a fantasy for a minute. Imagine a group of bright college freshmen, interested in mathematics for its own sake, with a solid grounding in high school mathematics. They will be your students for the next two or three years, and your job is to lead them through calculus and into the beginnings of higher analysis - complex variables and Fourier series, for example. You can present the material in any way you want, in any order you want. How would you proceed? Most of us would not be very creative in answering this question. ... It is therefore refreshing to contemplate the radically different overview of the subject in Roger Godement's four-volume 'Analyse Mathematique'. Based on his many years of teaching but written only after he retired, it is a worthy addition to the grand French tradition of the 'Cours d'Analyse'.

A later part of this review is given under Gerald Folland's review of Vol. II.

**6.2. Review by: D H Armitage.**

*Mathematical Reviews*, MR1671443

**(2000k:00004)**.

The volume under review is the first of three. Volumes I and II treat functions of real or complex variables, and Volume III will deal with analytic functions and the theory of integration. The book is written for readers who are interested in mathematics for its own sake. It does not treat applications, nor is its approach dictated by any programme of study prescribed for French universities. The author's style is very discursive, and there are many pithy remarks, not all directly to do with mathematics. The history of an idea is often presented in some detail with a critical analysis and comments about the mathematicians involved and the mathematical culture of their period. Thus the reader is led to an appreciation of how a particular theorem emerged, how it is related to other results, and which features of its proof merit special attention. The work will be of great interest even to readers who are already familiar with most of its mathematical content.

**6.3. Review by: Jean L Mawhin.**

Zentralblatt MATH (Zbl 0908.26001).

This is the first of the two volumes (for a review of the second volume see the following review) of a course of mathematical analysis taught by Roger Godement during thirty-five years at the University of Paris. The content is quite classical: sets and functions, convergence of sequences and series, continuous and differentiable functions, elementary functions. The treatment is less classical: precise although unpedantic (rather far from the style definition-theorem-corollary), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. The ones on set theory, real numbers, harmonic series, uniform convergence, Cauchy criterion, differentiable functions, logarithmic function, strange identities can be recommended. The author also gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction contains also comments which are very unusual in a book on mathematical analysis, going from pedagogy to critics of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that the reader is less surprised than he should.

**6.4. Review by: Gerald B Folland.**

*SIAM Review*

**47**(3) (2005), 602.

This is a review of the English translation

*Analysis I: Convergence, Elementary Functions*(2004).

Analysis I is the translation of the first volume of Godement's four-volume work 'Analyse Mathematique', which offers a development of analysis more or less from the beginning up to some rather advanced top ics. This volume begins with a short chapter on set theory and then proceeds to the development of various subjects that fall under the general rubric of "calculus." Although the material is almost all well within the scope of undergraduate courses, the style is that of a sophisticated mathematician writing for students who have a serious interest in theoretical mathematics. The writing is very personal and discursive: the author is telling an interesting story in his own voice with lots of asides and digressions, rather than constructing a formal edifice. ...

**6.5. Review by: Nick Lord.**

*The Mathematical Gazette*

**89**(514) (2005), 152-153.

This is a review of the English translation

*Analysis I: Convergence, Elementary Functions*(2004).

Amongst French mathematicians, there is a rich tradition of multi-volume Cours d'Analyse ranging from those of a century ago associated with the names of Jordan, Picard and Goursat to Dieudonne's more recent Treatise on analysis. In this tradition, Analysis I is an English translation of the first volume of a four-volume work. Although Godement (like Dieudonne) was a member of the author-collective Bourbaki, he here deliberately eschews the rigid, formal presentation associated with Bourbaki in favour of a leisurely, discursive style. This gives the text rather an old-fashioned feel; I think that readers will be split on whether or not Godement has been over-indulged by his editors in terms of the amount of commentary of a personal nature he has included. This ranges from witty, sometimes vitriolic, asides to longer, rather polemical paragraphs outlining Godement's world-view, especially his deeply-held concerns about modem weaponry and freedom of information. ... Although the content is 'elementary', there are several reasons why I do not think this is an introductory book. There are no exercises (other than of the 'complete the details' type) and beginners will find it quite hard to sift the key results from the commentary in order to navigate a route through the book. It is much more likely to find a resonance with those thoroughly familiar with the material who will respect Godement's lifetime of reflection on the material and fully appreciate his more teasing remarks.

**7. Analyse Mathématique II. Calcul différentiel et intégral, séries de Fourier, fonctions holomorphes (1998), by Roger Godement.**

**7.1. Review by: Gerald B Folland.**

*Amer. Math. Monthly*

**114**(2) (2007), 172-176.

This is a review of the English translation

*Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions*(2005).

An earlier part of this review is given under Folland's review of Vol. I.

... how might teachers of mathematics use Godement's Analysis, apart from their own entertainment and instruction? These books are not designed to be a text in any conventional sense, they have practically no exercises as such, and their unorthodox ordering of topics will make them difficult to fit into most curricula. Nonetheless, I think they can be of real value as supplementary reading for honours calculus and analysis courses. Various sections could serve as the basis for interesting individual projects. Simply browsing through the books will introduce the students to new perspectives, give them an unusual tour of the subject with an old master as guide, and point them toward the pleasures of more advanced mathematics. If Godement's digressions on the dark side of the Force give them a jolt now and then, so much the better. And if they experience the same kind of perplexity with the attitude in those digressions as I did, send them off to read Isaac Asimov's double-edged parable "The Feeling of Power", where they can find a neat demonstration of the perils not only of selling one's intellect to the military but of outsourcing it to an electronic device held in the palm of one's hand.

**7.2. Review by: Jean L Mawhin.**

Zentralblatt MATH (Zbl 0908.26002).

This second volume of Godement's

*Analyse mathématique*is devoted to integral calculus (Riemann integral with glimpses on Lebesgue integral, Radon measure and Schwartz distributions), asymptotic expansions, harmonic analysis and holomorphic functions. The style is similar to that of volume I, and the book concludes with a polemic postface of almost one hundred pages on

*Science, technology and weapons*, a mixture of generous ideas and local French politics, built around the famous discussion of Fourier and Jacobi about applied and pure mathematics. In contrast to the always appreciated scientific quotations, some of those occurring in this postface and throughout the book may be less appreciated. The reading of this book is recommended to mathematicians both for the inspiring style and taste of the presentation of the topics and for the unusual character of the comments: we learn, among deeper things, that Marshall Stone not only gave a new proof of Weierstrass, but had a definite taste for French gastronomy, and that Marcel Riesz was not only Frederic's brother, but had a strong taste for aquavit. This book, a definitive testimony of fact that mathematics is before all a human science, is an excellent antidote to the industrial character of a large part of the production of mathematical monographs.

**7.3. Review by: D H Armitage.**

*Mathematical Reviews*, MR1681199

**(2000k:00005)**.

This volume contains three chapters on real and complex analysis, which continue the work begun in Volume I, and a postscript on science, technology and armament. Again, the presentation of the mathematical component of the book is discursive, partly historical, and full of interest.

**8. Analyse Mathématique III. Fonctions analytiques, différentielles et variétés, surfaces de Riemann (2002), by Roger Godement.**

**8.1. Review by: Jean L Mawhin.**

Zentralblatt MATH (Zbl 0987.30001)

This is the third volume of the course of mathematical analysis taught by Roger Godement at the University of Paris. The first two volumes have already been analyzed, and the third one is written in the same spirit. Besides the technical aspects, written in a careful and luminous style, the reader will find many historical and personal remarks, including a defense of the role of Bourbaki in reply of some remarks of B Mandelbrot, and a comparison of the "proofs" of the Stokes theorem by physicists and mathematicians. The first topics treated in this volume is Cauchy's theory of holomorphic functions, including a very careful treatment of the integral theorems, and detailed applications to the real and complex Fourier transforms, gamma function, Hankel integral, Mellin transform and Dirichlet problem on a half-plane. The next chapter develops the calculus of functions of several variables, tensor calculus, differential forms, differential manifolds and Stokes theorem. It includes a detailed treatment of the formula of change of variables in a multiple integral. The last chapter is devoted to a detailed treatment of the Riemann surface of an algebraic function. It ends with a very vivid description of the algebraic viewpoint. The pleasure and interest in reading such a book is the same as for the first two volumes, and one is looking forward discovering the fourth and last one.

**8.2. Review by: P Lappan.**

*Mathematical Reviews*, MR2164651

**(2006i:30001)**.

This is the third volume of the author's extensive treatise on analysis. Although the order of topics follows no standard curriculum, the combined volumes give a detailed treatment of real analysis and complex analysis. In each section, the book has the feel of a very careful textbook, where each claim is proved in complete detail. While the author skips back and forth between real and complex analysis, there seems to be an attempt to cycle back over important ideas, adding a slightly deeper layer each time. In the third volume, the author both expands on some of the topics treated in the first two volumes, providing substantial generalization, and also introduces many new topics. In addition, there are a number of historical and philosophical asides. While there are some problems for students, the number of such problems is not sufficient for adequate student practice and/or extensions of the material, and thus some supplementary problems would be desirable if portions of the book were to be used as a course textbook. The focus of this volume is on some topics in complex analysis, especially integral representations and their consequences, and the differential calculus of varieties. ... The book is well written and mathematically complete, with many explanations of the basic mathematical ideas in non-technical language combined with the precise mathematical formulations.

**9. Analyse Mathématique IV. Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires (2003), by Roger Godement.**

**9.1. Review by: Solomon Friedberg.**

*Mathematical Reviews*, MR1995794

**(2005k:11002)**.

The volume under review, consisting of two chapters numbered XI and XII, is the fourth in a series. The first three volumes treat functions of real and complex variables. The first chapter of this volume concerns integration, spectral theory, and harmonic analysis; the second concerns modular forms and related topics. ... This book is written with a particular and engaging style, as described in the reviews of the previous volumes (see, e.g., the review of the first volume). Its reader will be rewarded with a sophisticated and tasteful perspective on the topics under consideration, coupled with an absorbing collection of historical and personal remarks and observations.