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# Maurice Fréchet

Maurice Fréchet (September 2, 1878 – June 4, 1973) was a French mathematician. He made major contributions to the topology of point sets and introduced the entire concept of metric spaces. He also made several important contributions to the field of statistics and probability, as well as calculus. His dissertation opened the entire field of functionals on metric spaces and introduced the notion of compactness. Independently of Riesz, he discovered the representation theorem in the space of Lebesgue square integrable functions.

Biography

Early life

He was born to a Protestant family in Maligny, Yonne to Jacques and Zoé Fréchet. At the time of his birth, his father was a director of a Protestant orphanage in Maligny and was later in his youth appointed a head of a Protestant school. However, the newly established Third Republic was not sympathetic to religious education and so the laws were enacted requiring all education to be secular. As a result, his father lost his job. To generate some income his mother set up a boarding house for foreigners in Paris. His father was able later to obtain another teaching position within the secular system - it was not a job of a headship, however, and the family could not expect as high standards as they might have otherwise.

Maurice attended the secondary school Lycée Buffon in Paris where he was taught mathematics by Jacques Hadamard. Hadamard recognised the potential of young Maurice and decided to tutor him on an individual basis. After Hadamard moved to the University of Bordeaux in 1894, Hadamard continuously wrote to Fréchet, setting him mathematical problems and harshly criticising his errors. Much later Fréchet admitted that the problems caused him to live in a continual fear of not being able to solve some of them, even though he was very grateful for the special relationship with Hadamard he was privileged to enjoy.

After completing high-school Fréchet was required to enroll in the military service. This is the time when he was deciding whether to study mathematics or physics - he chose mathematics out of dislike of chemistry classes he would have had to take otherwise. Thus in 1900 he enrolled to École Normale Supérieure to study mathematics.

He started publishing quite early, having published four papers in 1903. He also published some of his early papers in the American Mathematical Society due to his contact with American mathematicians in Paris—particularly Edwin Wilson.

Middle life

Fréchet served at many different institutions during his academic career. From 1907-1908 he served as a professor of mathematics at the Lycée in Besançon, then moved in 1908 to the Lycée in Nantes to stay there for a year. After that he served at the University of Poitiers between 1910-1919.

He married in 1908 to Suzanne Carive and had four children: Hélène, Henri, Denise and Alain.

First World War

Fréchet was planning to spend a year in the United States at the University of Illinois but his plan was disrupted when the First World War broke out in 1914. He was mobilised on 4 August the same year. Because of his diverse language skills, gained when his mother ran the establishment for foreigners, he served as an interpreter for the British Army. However, this was not a safe job and he spent two and a half years very near or at the front. The French egalitarian ideas meant that very many academics were mobilised to serve in the trenches and many of them were lost during the war. It is remarkable that during his service in the war he still managed to produce frequent cutting edge mathematical papers, even with very little time available to devote to mathematics.

After the war

After the end of the war, Fréchet was chosen to go to Strasbourg to help with the reestablishment of the university. He served as a professor of higher analysis and Director of the Mathematics Institute. Despite being burdened with administrative work, he was again able to produce a large amount of high quality research.

In 1928 Fréchet decided, with encouragement from Borel, to move back to Paris. He held various positions in Paris until his retirement in 1948. Despite his major achievements, he was not overly appreciated in France. As an illustration, while being nominated numerous times, he was not elected a member of the Academy of Sciences until the age of 78.

Fréchet was an Esperantist, publishing some papers and articles in the international language.[1] He also served as president of the Internacia Scienca Asocio Esperantista ("International Scientific Esperantist Association") from 1950-53.[2]

Main works

His first major work was his outstanding PhD thesis that he submitted in 1906. The thesis was titled Sur quelques points du calcul fonctionnel and was concerned with the calculus of functionals. Here Fréchet introduced the concept of a metric space, although the name is due to Hausdorff. The level of abstraction used by Fréchet is similar to that in group theory, allowing one to develop an axiomatic system to study a large array of mathematical objects. The system can then be applied in very many particular cases. Fréchet[3] developed here ideas from the article Deux types fondamentaux de distribution statistique[4] (1938; an English translation Two Basic Types of Statistical Distribution) of Czech geographer, demographer and statistician Jaromír Korčák.

Here is a list of his most important works, in chronological order:

* Sur les opérations linéaires I-III, 1904-1907 (On linear operators)

* Les Espaces abstraits, 1928 (Abstract spaces)

* Recherches théoriques modernes sur la théorie des probabilités, 1937-1938 (Modern theoretical research in the theory of probability)

* Les Probabilités associées à un système d'événements compatibles et dépendants, 1939-1943 (The Probabilities Associated with a System of Compatible and Dependent Events)

* Pages choisies d'analyse générale, 1953 (Selected Pages of General Analysis)

* Les Mathématiques et le concret, 1955 (Mathematics and the concrete)

See also

* Fréchet space

* Fréchet-Urysohn space

* Fréchet derivative

* Differentiation in Fréchet spaces

* Topological space

* Functional analysis

* Fréchet distribution

* Fréchet mean

* Fréchet distance

* Fréchet filter

Notes

1. ^ La kanonaj formoj de la 2, 3, 4-dimensiaj paraanalitikaj funkcioj (Esperanto)

2. ^ from the Esperanto Wikipedia

3. ^ Maurice R. Fréchet (1941): Sur la loi de répartion de certaines grandeurs geographiques. Journal de la societé de Statistique du Paris, 82, s. 114-122.

4. ^ Jaromír Korčák (1938): Deux types fondamentaux de distribution statistique. Prague, Comité d’organisation, Bull. de l'Institute Int'l de Statistique, vol. 3, pp. 295–299.

External links

* O'Connor, John J.; Robertson, Edmund F., "Maurice René Fréchet", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Frechet.html .

* Maurice René Fréchet at the Mathematics Genealogy Project

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