# René Maurice Fréchet

### Born: 2 September 1878 in Maligny, Yonne, Bourgogne, France

Died: 4 June 1973 in Paris, France

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**Maurice Fréchet**'s parents were Jacques and Zoé Fréchet. It was a Protestant family and at the time Maurice was born his father was the director of a Protestant orphanage in Maligny. There were six children in the family, Maurice being the fourth. While he was still a very young child, his father Jacques Fréchet was appointed as head of a Protestant school in Paris and the family moved there with high expectations of a good future. However, political decisions were to severely damage the comfortable life of the Fréchet family.

Before Maurice was born the constitution of the Third Republic had been drawn up. Although the Republicans were divided into two wings, these were united on their attitude to the church's role in politics and education. One of the leaders Jules Ferry held major positions of power during 1880-85 and he brought in new laws to make primary education free, compulsory, and secular. Religious teaching in schools was replaced by "civic education" and from this time on French education was secularised. Jacques Fréchet lost his job as headmaster and was unemployed. Maurice's mother came to the financial rescue of the family by setting up a boarding house for foreigners. This had the benefit that Maurice grew up surrounded by those speaking foreign languages and he developed an international outlook which was to remain with him throughout his life. After the French education system settled down from the impact of the legal restraints put upon it, Jacques Fréchet was able to find a job teaching within the new secular system. It meant that the family never enjoyed the standards they might otherwise have expected, but nevertheless they did return to a stable financial situation.

Maurice entered secondary education at the Lycée Buffon in Paris. There he was taught mathematics by Hadamard who was a teacher at the school from 1890 to 1893 before being appointed professor at the University of Bordeaux in 1894. Hadamard immediately saw the mathematical potential of his young pupil and coached him on an individual basis. This continued after Hadamard moved to Bordeaux, for he wrote to Fréchet setting him mathematical problems, and corrected his work with severe criticisms if there were any errors. The relationship was one in which Fréchet was extremely grateful for the encouragement and guidance that he was receiving, but he admitted much later that he lived in continual fear of not being able to solve the problems he was set.

After leaving school, Fréchet undertook military service before, in 1900, entering the École Normale Supérieure in Paris. There he still worried over the decision on whether to specialise in physics or mathematics, and was eventually persuaded to specialise in mathematics, not because he did not enjoy physics just as much, but rather because further study of physics required him to take chemistry courses which he disliked. Even before he was awarded his Agrégation des Sciences Mathematiques in 1903, Fréchet began publishing short papers. By the end of 1903 he had four papers in print, three of which were four pages long. Seven further papers appeared in 1904, then remarkably eleven papers in 1905 as he undertook research for his doctorate under Hadamard's supervision. Contact with several American mathematicians who were in Paris, in particular Edwin Wilson, led to Fréchet publishing some of his early papers in American Mathematical Society publications (Edwin Wilson was editor of the

*Transactions of the American Mathematical Society*from 1903). Another task undertaken by Fréchet around this time was writing up Borel's lectures for publication. Fréchet attended these lectures while an undergraduate and wrote up the lectures during the winter of 1903-04. The book

*Leçons sur les fonctions de variables réelles et les développements en séries de polynômes*Ⓣ was published in 1905.

Fréchet wrote an outstanding doctoral dissertation

*Sur quelques points du calcul fonctionnel*Ⓣ submitted on 2 April 1906. In it he introduced the concept of a metric space, although he did not invent the name 'metric space' which is due to Hausdorff. The thesis concerns 'functional operations' and 'functional calculus' and is developed from ideas due to Hadamard and Volterra. The importance of the thesis is that it develops axiomatic analysis systems providing an abstraction of different objects studied by analysis in a similar way to group theory providing an abstraction of algebraic systems. This parallel is drawn by Fréchet himself who requires sufficient structure on his abstract systems so that limits and continuity can be studied. He defines a functional operation as a numerically valued function defined on arbitrary objects which he wants to include points, lines, functions, numbers, surfaces etc. The functional calculus of his thesis is then the systematic study of functional operations.

A versatile mathematician, Fréchet served as professor of mathematics at the Lycée in Besançon (1907-08), professor of mathematics at the Lycée in Nantes (1908-09), then professor of mechanics at the Faculty of Science in Poitiers (1910-19). He married Suzanne Carrive in 1908 and they had four children; Hélène, Henri, Denise, and Alain. Fréchet had arranged to spend the academic year 1914-15 at the University of Illinois at Urbana in the United States and had accepted an appointment there for one year. He and his family were packed and ready to travel to the port to board their ship for the United States when World War I broke out and Fréchet was required for military service.

He was mobilised on 4 August 1914 but because of his language skills, initially gained when his mother ran the establishment of foreigners, he was attached to the British Army as an interpreter. This may have resulted in a slightly safer job than he would otherwise have had, but nevertheless he spent about two and a half years at or near the front, so was fortunate to survive. A great many French academics perished during the war, for the French belief in equality meant they tended to fight in the trenches rather than undertake specialised war work for which their expertise made them especially useful. In fact there is evidence that Fréchet had arranged with some American mathematicians to publish his complete works if he did not survive the war. That he chose to negotiate with Americans is almost certainly a sign that he felt more appreciated in that country than in his own. Not only did he undertake this dangerous work during the war, but Fréchet continued to produce frequent mathematics papers. One can only marvel at how he was able to continue with cutting edge research in such circumstances and with so little time to devote to his mathematics.

For the period of the war Fréchet retained his post at the Faculty of Science in Poitiers despite not being able to teach there. However before he was released from military service at the end of the war, he was selected to go to Strasbourg to assist with re-establishing the university there. He was both professor of higher analysis at the University of Strasbourg and Director of the Mathematics Institute there from 1919 to 1927. As he had been in earlier times, Fréchet was able to continue to produce a large research output despite heavy duties. He now had major administrative duties, one of the first being setting up and organising the International Congress of Mathematicians in Strasbourg in 1920. This was a difficult Congress for political reasons, since German and Austrian mathematicians were banned but this resulted in strong opinions and numerous arguments. An indication of his remarkable research output is that he had 36 papers published in the two years 1924 and 1925. It was after going to Strasbourg that he began to become interested in statistics but he only published a small number of articles on probability at this stage, most of his papers being on general analysis and topology. However, he taught courses on probability, statistics, and insurance mathematics at Strasbourg.

From November 1928 Fréchet held posts in Paris, but from this time on he concentrated more on statistics. It was Borel who encouraged Fréchet to seek positions in Paris and he supported his candidacy. There is also a suggestion that Fréchet had a difference of opinion with the Council of the Faculty of Science at Strasbourg which meant he was both pleased to return to Paris and not unhappy at leaving Strasbourg. He held several different positions in the field of mathematics in Paris between 1928 and 1948 when he retired. He was director of studies at the École des Hautes-Études, then professor at the Faculty of Science in Paris. From 1929 he was also professor of analysis and mechanics at the École Normale Supérieure. In [6] Armatte examines in detail the:-

In January and February of 1942 Fréchet was lecturing in Portugal. He lectured in Lisbon on:... campaign[Fréchet]sustained from1934to1936at the International Institute of Statistics against improper uses of the correlation coefficient. This campaign took the unusual form of a survey sent to colleagues all over the world as well as a series of papers, committee reports, and censure motions within the International Institute of Statistics. It sheds light on the difficulties of giving a mathematically sound foundation to one of the most elementary notions of mathematical statistics, at the key moment when it was developing into an autonomous scientific discipline. Moreover, correlation is a notion largely used as an instrument of proof in several domains of observational sciences. Maurice Fréchet did not disdain contributing his own stone to the building of these mathematical foundations via his work on the notion of distance. And, because he respected applied mathematics, he used the same surveying technique - and with the same pugnacity - to examine the consistency and relevance of other statistical methods, like the estimation of the parameters of a theoretical distribution and the application of mathematics to economic and social questions.

*Les fonctions périodiques, les fonctions presque périodiques et les fonctions assymptotiquement presque périodiques; Applications des fonctions assymptotiquement presque périodiques au théorème ergodique de Birkhoff; Les débuts de la topologie combinatoire; le théorème d'Euler-Cauchy; La théorie des courbes dans les espaces abstraits très généraux; Types homogènes de dimensions*Ⓣ; and

*Le développement d'une fonction continue en série de polynômes dans les espaces abstraits*Ⓣ. In addition he gave a lecture aimed at the general public on

*Les origines des notions mathématiques*Ⓣ.

The Portuguese Mathematical Society made Fréchet an honorary member and recognised four major achievements:

- The deep influence exerted by the ideas of Maurice Fréchet in contemporary mathematics.

- That the creation of a theory of abstract spaces, by this illustrious mathematician, at the beginning of the century, was the necessary starting point for the development of numerous mathematical theories which flourish today.

- That the abstract theories of which Maurice Fréchet was one of the first pioneers, have allowed a major work of synthesis and clarification in the mathematical sciences to be carried out through the last thirty years.

- That the workmanship which Maurice Fréchet carried out in the most varied fields of the pure and applied mathematics is a monument to the glory of the constructive spirit of the Man.

Fréchet's most important work includes:

1.

3. (with Ky Fan)

Finally let us mention that Fréchet was an extremely active correspondent with most of the leading mathematicians of his day. Let us record just a few of the names: Pavel Sergeevich Aleksandrov, René-Louis Baire, L E J Brouwer, Béla Kerékjártó, Kazimierz Kuratowski, Henri Lebesgue, Paul Lévy, Nikolai Nikolaevich Luzin, P Mahlo, Paul Montel, Frigyes Riesz, Wacław Sierpiński, Pavel Samuilovich Urysohn, Edwin Wilson, and Stanisław Zaremba. As an example of one of these correspondences we refer to [4] which is described by Esther R Phillips as follows:-
*Les Espaces abstrait*Ⓣ (1928), in which he:-2.... made a major contribution toward laying the foundations of general topology and abstract analysis.

*Récherchés théoretiques modernes sur la théorie des probabilités*Ⓣ (1937-38).3. (with Ky Fan)

*Introduction à la Topologie Combinatoire*Ⓣ (1946), about which Eilenberg writes:-4.Entertaining reading about combinatorial topology accessible to a reader with very little mathematical preparation. The topics discussed are: the Jordan curve theorem, the map colouring problem, the Euler characteristic and the classification of surfaces. The proofs are very intuitive and are not intended to be complete. Many historical remarks are included.

*Pages choisies d'analyse générale*Ⓣ (1953), which:-5.... contains selections by[Fréchet]from his papers on general analysis. He has also supplemented the original papers with short remarks on later developments connected with the ideas in them, answers to conjectures, etc. The papers are grouped under the following headings: Vue d'ensemble; Espaces fonctionnels; Analyse fonctionnelle; Les espaces abstraits; L'analyse généraleⓉ.

*Les Mathématiques et le concret*Ⓣ (1955),This is a collection of a number of papers by Fréchet grouped under the general headings: Sur les mathématiques en général; Sur le calcul des probabilités et ses applications; Les mathématiciens et la vie. Deux examples. Papers have been selected for their general interest and readability by non-mathematicians.

Maurice Fréchet, according to the author, corresponded with virtually every mathematician of this century. His prodigious correspondence, preserved in the Archive of the Paris Academy of Sciences, sheds light on the genesis and evolution of a broad body of contemporary mathematics. Among Fréchet's manuscripts are forty-eight letters of which seven were written by P S Aleksandrov and P S Urysohn and the remainder (after Urysohn's death in 1924) by Aleksandrov. Of particular interest in connection with the development of topology are the letters written between 1920 and the 1930s. In the earliest of these letters the young Russian scholars express their gratitude to Fréchet for having created the theory of abstract spaces on which their earliest investigations were based and cite as the source of their first published works problems posed by N Luzin in his analysis seminar at Moscow University. The letters reveal that after Urysohn's death, Aleksandrov's investigations moved away from the "classification of topological spaces" and became increasingly algebraic [combinatorial], culminating in his simplicial theory of homology.

Although widely honoured, it does appear that Fréchet was more highly rated outside France than inside it. He was invited to address the International Congress of Mathematicians in Bologna in 1928 and in Oslo in 1932. He was elected to the Polish Academy of Sciences in 1929 and the Royal Society of Edinburgh in 1947. He was also a member of the International Institute of Statistics. He lost out many times in elections to the Academy of Sciences, eventually being elected in 1956 when he was 78 years old.

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

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**List of References** (22 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

**Honours awarded to Maurice Fréchet**

(Click the link below for those honoured in this way)

1. | Speaker at International Congress | 1928 | |

2. | Speaker at International Congress | 1936 | |

3. | Honorary Fellow of the Edinburgh Maths Society | 1938 | |

4. | Fellow of the Royal Society of Edinburgh | 1947 |

**Cross-references in MacTutor**

- History Topics: Topology enters mathematics
- Chronology: 1900 to 1910
- French Mathematical Society
- Portuguese Mathematical Society
- 1912 ICM - Cambridge
- 1924 ICM - Toronto
- 1928 ICM - Bologna
- 1936 ICM - Oslo

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