**Jean Dieudonné**'s father was Ernest Dieudonné, who was an industrialist, and his mother was Léontine Labrun. As a young child Jean was irresistibly attracted to dictionaries, encyclopaedias, and universal histories. He studied at the Lycée in Lille where his love of mathematics flourished by age fourteen when he began to study algebra. After completing his school studies he entered the École Normale Supérieure in Paris where he was inspired by Émile Picard, Jacques Hadamard, Élie Cartan, Paul Montel, Arnaud Denjoy and Gaston Julia. Dieudonné received both his bachelor's degree (1927) and his doctorate (1931) from the École Normale. His doctoral studied were supervised by Montel and his thesis was in the area of classical analysis. He worked at the Faculty of Science at Rennes as Maître de Conférences from 1933, and on 22 July 1935 he married Odette Clavel; they had two children Jean-Pierre and Françoise. By the time of his marriage he had already become one of the founding members of Bourbaki. It greatly changed his mathematical outlook. He wrote:-

Left to myself, I would undoubtedly have remained billeted in a narrow sector of analysis my whole life.

He worked, also as Maître de Conférences, at the Faculty of Science at Nancy from 1937 to 1946. He was then appointed professor of mathematics at São Paulo in Brazil (1946-47). Returning to Nancy, he was professor there in the Faculty of Science from 1948 to 1952 when he accepted a one year appointment as professor of mathematics at the University of Michigan. After teaching at Northwestern University from 1953 to 1959 Dieudonné returned to France to take up an appointment as professor of mathematics at the Institut des Hautes Études Scientifiques. In 1964, after five years at the Institut des Hautes Études Scientifiques, he accepted a chair at the faculty of Science at Nice, a post which he held until 1970.

We note that Dieudonné's love was for research and not for teaching. E Beckenstein, reviewing Dieudonné's *Choix d'oeuvres mathématiques* Ⓣ, writes:-

He never had the least inclination to teaching. Its only attractive feature was that it provided him with enough time to pursue his own research, an opportunity that fate did not grant to men of the stature of Kummer, Weierstrass, Grassmann, Killing or Montel, who spent most of their careers in the far more time-consuming endeavour of secondary teaching. Even after forty years of teaching, he was still more at ease in front of a sheet of paper than a listener. He always used notes when he lectured "pour éviter les catastrophes". He makes no apologies for being a typical ivory tower type: only by incessant thought can some things be achieved. But ivory tower notwithstanding, he is not an ascetic, never having disdained the pleasures of existence.

We mentioned above that Dieudonné was a founder member of Bourbaki. He was one of the main contributors to the Bourbaki series of texts from the time that the group came into existence and in many ways he was the leading influence in a group whose whole object was to avoid anyone taking on this role. Speaking of Bourbaki congresses, which he loved, Dieudonné writes in [5]:-

Certain foreigners, invited as spectators to Bourbaki meetings, always come out with the impression that it is a gathering of madmen. They could not imagine how these people, shouting -- some times three or four at the same time -- could ever come up with something intelligent...

Speaking of his own involvement in Bourbaki and its influence on his own career, Dieudonné writes [5]:-

In my personal experience, I believe that if I had not been submitted to this obligation to draft questions I did not know a thing about, and to manage to pull through, I should never have done a quarter or even a tenth of the mathematics I have done.

He began his mathematical career working on the analysis of polynomials. He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups.

His best known books are *La Géométrie des groupes classiques* Ⓣ (1955), *Foundations of Modern Analysis* (1960), *Algèbre linéaire et géométrie élémentaire* Ⓣ (1964) and nine volumes of *Éléments d'analyse* Ⓣ (1960-1982). C E Rickart, reviewing *La Géométrie des groupes classiques* Ⓣ writes:-

This volume brings together most of the modern results concerning the so-called elementary theory of the classical groups. Here the term "classical group" is used as in the author's monograph, Sur les groupes classiques(1948)and the "elementary theory" refers roughly to results which involve subgroup and homomorphisms as opposed to results concerned for example with topology, differential geometry, etc. The approach is, of course, algebraic but, as is characteristic of the author's work in this field, is strongly influenced by geometrical notions. Although many mathematicians have contributed to the subject, the bulk of the results presented here are due to the author, the main references being the monograph cited above and his paper, On the automorphisms of the classical group(1951).

Dieudonné writes in *Foundations of Modern Analysis* that it is intended:-

...to provide the necessary elementary background for all branches of modern mathematics involving 'analysis'.

J L Kelly writes:-

The most remarkable feature of the text is the consistently geometrical formulation of the results. For example, the differential calculus is developed in terms of linear approximation to functions on an open subset of a Banach space to a Banach space. Yet it would be completely false to assert that the book contained a study of Banach spaces - no non-trivial proposition on such spaces is proved. The subject of study is indeed elementary analysis, and the theorems are theorems of analysis stated in geometrical terms. This geometrization is rather like the geometrization of linear algebra which occurred some years ago, and, as in the linear algebra case, there are enormous conceptual and technical advantages. A good deal is accomplished in the350pages of the text. The mathematical organization is superb, the presentation lucid, there are a large number of very good problems, and there are excellent expository introductions to each chapter(couched in the author's customary diffident style). In brief, it is a beautiful text.

In writing *Algèbre linéaire et géométrie élémentaire* Ⓣ Dieudonné aims to provide teachers in the lycées of France with sufficient background in geometry so that they can prepare their pupils properly for entry to university study. He presents the topic in terms of linear (or geometric) algebra of two and three dimensions. Let us not pass judgement on whether the text is too sophisticated to fulfil its intended purpose but we do note that in introducing the real numbers in the first chapter Dieudonné assumes they are an ordered field in which the intermediate value theorem is valid for polynomials of degree 3.

We should also examine Dieudonné's contributions as a historian of mathematics. He published texts such as *History of functional analysis* (1981), *History of algebraic geometry* (1985), *Pour l'honneur de l'esprit humain* Ⓣ (1987), *A history of algebraic and differential topology *1900-1960 (1989), and *L'école mathématique française du XXe siècle* Ⓣ (2000).

The *History of functional analysis* is:-

... a detailed and absorbing account of the history and development of functional analysis, beginning with Lagrange and Daniel Bernoulli, through the work of Fredholm, Hilbert, and Frigyes Riesz at the turn of this century, and ending about1960.

Mac Lane, in a review of *A history of algebraic and differential topology*, writes:-

... is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincaré and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategy of its proofs and its relation to other work. Previously, the history of the many technical developments of20th-century mathematics had seemed to present insuperable obstacles to scholarship. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.

In addition to the historical texts, Dieudonné edited the works of Camille Jordan. In the first volume Dieudonné has contributed an article on Jordan's work on finite groups and in the second volume an interesting 116-page introduction to Jordan's work on linear and multilinear algebra and on the theory of numbers. Dieudonné also wrote a preface to the mathematical writings and memoirs of Galois which were published in 1962.

Several descriptions of Dieudonné, particularly of those involved with him in the Bourbaki project, are interesting. Armand Borel writes in [3]:-

For about twenty-five years he would routinely start his day(maybe after an hour of piano playing)by writing a few pages for Bourbaki. In particular, but by far not exclusively, he took care of the final drafts, exercises, and preparation for the printer of all the volumes(about thirty)which appeared while he was a member and even slightly beyond. This no doubt accounts to a large extent for the uniformity of style of the volumes, frustrating any effort to try to individualize one contribution or the other. But this was not really Dieudonné's style, rather the one he had adopted for Bourbaki.

Pierre Cartier said in [10]:-

Dieudonné was quite a good piano player, at an amateur level, but quite good, and he had a fantastic memory. He knew hundreds and hundreds of pages of score by heart and could follow every single note. I remember I had a few occasions to go to the concert hall with him. It was fascinating, he would look at the score in his hand and exclaim "OH!" if a note was missing from the orchestra! He devoted the last six months of his life - when he decided that his mathematical life was finished, he had written his last book, and he retreated to his home - to listening to recordings and following the scores and the notes.

When Dieudonné was the scribe of Bourbaki, for many many years, every printed word came from his pen. Of course there had been many drafts and preliminary versions, but the printed version was always from the pen of Dieudonné. And with his fantastic memory, he knew every single word. I remember, it was a joke, you could say, "Dieudonné, what is this result about so and so?" and he would go to the shelf and take down the book and open it to the right page.

We can obtain an appreciation of Diedonné's views of mathematics from a number of sources. First we quote Dieudonné's metaphorical ball of yarn from [5]:-

Here is my picture of mathematics now. It is a ball of wool, a tangled hank where all mathematics react upon another in an almost unpredictable way. And then in this ball of wool, there are a certain number of threads coming out in all directions and not connecting with anything else. Well the Bourbaki method is very simple-we cut the threads.

In [9] Medvedev quotes these words of Dieudonné written in a 1976 article:-

... the principal factor in the development of mathematics has an internal origin - reflection on the nature of the open problems, independently of their origin.

Dieudonné was elected to the Academy of Sciences (Paris) in 1968, received the Gaston Julia prize in 1966, and he was made an Officer of the Légion d'Honneur.

**Article by:** *J J O'Connor* and *E F Robertson*

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