René Maurice Fréchet
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René Maurice Fréchet (; 2 September 1878 – 4 June 1973) was a French mathematician. He made major contributions to
general topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
and was the first to define
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s. He also made several important contributions to the field of
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
and
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
, as well as
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
. His dissertation opened the entire field of functionals on
metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for ...
and introduced the notion of
compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
. Independently of
Riesz Riesz may refer to: * Frigyes Riesz (1880–1956), Hungarian mathematician * Marcel Riesz (1886–1969), Hungarian and Swedish mathematician, younger brother of Frigyes Riesz See also * Riesz, the fictional Amazon warrior and princess of Rolant ...
, he discovered the
representation theorem In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure. Examples Algebra * Cayley's theorem states that every group i ...
in the space of Lebesgue square integrable functions. He is often referred to as the founder of the theory of abstract spaces.


Biography


Early life

He was born to a
Protestant Protestantism is a branch of Christianity that emphasizes Justification (theology), justification of sinners Sola fide, through faith alone, the teaching that Salvation in Christianity, salvation comes by unmerited Grace in Christianity, divin ...
family in Maligny 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 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 The Lycée Buffon is a secondary school in the XVe arrondissement of Paris, bordered by boulevard Pasteur, the rue de Vaugirard and the rue de Staël. Its nearest métro station is Pasteur. It is named for Georges-Louis Leclerc, comte de Buffon ...
in Paris where he was taught mathematics by
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a tea ...
. Hadamard recognised the potential of young Maurice and decided to tutor him on an individual basis. After Hadamard moved to the
University of Bordeaux The University of Bordeaux (, ) is a public research university based in Nouvelle-Aquitaine in southwestern France. It has several campuses in the cities and towns of Bordeaux, Dax, Gradignan, Périgueux, Pessac, and Talence. There are al ...
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 military service. This is the time when he was deciding whether to study mathematics or physics – he chose mathematics out of dislike of the chemistry classes he would have had to take otherwise. Thus in 1900 he enrolled to
École Normale Supérieure École or Ecole may refer to: * an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing i ...
to study mathematics. He started publishing quite early, having published four papers in 1903. He also published some of his early papers with the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
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 to 1908 he served as a professor of mathematics at the Lycée in
Besançon Besançon (, ; , ; archaic ; ) is the capital of the Departments of France, department of Doubs in the region of Bourgogne-Franche-Comté. The city is located in Eastern France, close to the Jura Mountains and the border with Switzerland. Capi ...
, then moved in 1908 to the Lycée in
Nantes Nantes (, ; ; or ; ) is a city in the Loire-Atlantique department of France on the Loire, from the Atlantic Ocean, Atlantic coast. The city is the List of communes in France with over 20,000 inhabitants, sixth largest in France, with a pop ...
to stay there for a year. After that he served at the
University of Poitiers The University of Poitiers (UP; , ) is a public university located in Poitiers, France. It is a member of the Coimbra Group. It is multidisciplinary and contributes to making Poitiers the city with the highest student/inhabitant ratio in France ...
between 1910 and 1919. He married in 1908 to Suzanne Carrive (1881–1945) 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 The University of Illinois Urbana-Champaign (UIUC, U of I, Illinois, or University of Illinois) is a public university, public land-grant university, land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United ...
, but his plan was disrupted when the
First World War World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...
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 The British Army is the principal Army, land warfare force of the United Kingdom. the British Army comprises 73,847 regular full-time personnel, 4,127 Brigade of Gurkhas, Gurkhas, 25,742 Army Reserve (United Kingdom), volunteer reserve perso ...
. However, this was not a safe job; he spent two and a half years very near to or at the front. French egalitarian ideals caused many academics to be mobilised. They served 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 cutting edge mathematical papers frequently, despite having little time to devote to mathematics.


After the war

After the end of the war, Fréchet was chosen to go to
Strasbourg Strasbourg ( , ; ; ) is the Prefectures in France, prefecture and largest city of the Grand Est Regions of France, region of Geography of France, eastern France, in the historic region of Alsace. It is the prefecture of the Bas-Rhin Departmen ...
to help with the reestablishment of the
university A university () is an educational institution, institution of tertiary education and research which awards academic degrees in several Discipline (academia), academic disciplines. ''University'' is derived from the Latin phrase , which roughly ...
. 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 to move back to Paris, thanks to encouragement from
Borel Borel may refer to: People * Antoine Borel (1840–1915), a Swiss-born American businessman * Armand Borel (1923–2003), a Swiss mathematician * Borel (author), 18th-century French playwright * Borel (1906–1967), pseudonym of the French actor ...
, who was then chair in the Calculus of Probabilities and Mathematical Physics at the Sorbonne. Fréchet briefly held a position of lecturer at the Sorbonne's
Rockefeller Foundation The Rockefeller Foundation is an American private foundation and philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The foundation was created by Standard Oil magnate John D. Rockefeller (" ...
and from 1928 was a Professor (without a chair). Fréchet was promoted to
tenured Tenure is a type of academic appointment that protects its holder from being fired or laid off except for cause, or under extraordinary circumstances such as financial exigency or program discontinuation. Academic tenure originated in the United ...
Chair of General Mathematics in 1933 and to Chair of Differential and Integral Calculus in 1935. In 1941 Fréchet succeeded Borel as chair in the Calculus of Probabilities and Mathematical Physics, a position Fréchet held until he retired in 1949. From 1928 to 1935 Fréchet was also put in charge of lectures at the
École Normale Supérieure École or Ecole may refer to: * an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing i ...
; in this latter capacity Fréchet was able to direct a significant number of young mathematicians toward research in probability, including Doeblin, Fortet, Loève, and Jean Ville. Despite his major achievements, Fréchet was not overly appreciated in France. As an illustration, while being nominated numerous times, he was not elected a member of the
Académie des sciences The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...
until the age of 78. In 1929 he became foreign member of the
Polish Academy of Arts and Sciences The Polish Academy of Arts and Sciences or Polish Academy of Learning (, PAU), headquartered in Kraków and founded in 1872, is one of two institutions in contemporary Poland having the nature of an academy of sciences (the other being the Po ...
and in 1950 foreign member of the
Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences (, KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed in the Trippenhuis in Amsterdam. In addition to various advisory a ...
. Fréchet was an
Esperantist An Esperantist () is a person who speaks, reads or writes Esperanto. According to the Declaration of Boulogne, a document agreed upon at the first World Esperanto Congress in 1905, an Esperantist is someone who speaks Esperanto and uses it for ...
, publishing some papers and articles in that constructed language. He also served as president of the ''Internacia Scienca Asocio Esperantista'' ("International Scientific Esperantist Association") from 1950 to 1953.


Main works

His first major work was his outstanding 1906 PhD thesis ''Sur quelques points du calcul fonctionnel'', on the calculus of functionals. Here Fréchet introduced the concept of a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, although the name is due to Hausdorff. Fréchet's level of abstraction is similar to that in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, proving theorems within a carefully chosen axiomatic system that can then be applied to a large array of particular cases. 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 (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) Fréchet also developed ideas from the article ''Deux types fondamentaux de distribution statistique'' (1938; an English translation ''The Two Fundamental Types of Statistical Distribution'') of Czech
geographer A geographer is a physical scientist, social scientist or humanist whose area of study is geography, the study of Earth's natural environment and human society, including how society and nature interacts. The Greek prefix "geo" means "earth" a ...
,
demographer Demography () is the statistical study of human populations: their size, composition (e.g., ethnic group, age), and how they change through the interplay of fertility (births), mortality (deaths), and migration. Demographic analysis examine ...
and statistician Jaromír Korčák. Fréchet is sometimes credited with the introduction of what is now known as the
Cramér–Rao bound In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic (fixed, though unknown) parameter. The result is named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, but has also been d ...
, but Fréchet's 1940s lecture notes on the topic appear to have been lost.


Family

In 1908 he married Suzanne Carrive.


See also

*
Differentiation in Fréchet spaces In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is Gateaux derivative between Fréchet spaces, i ...
*
F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
*
Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued f ...
*
Fréchet distance In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet. Intuitive definition Imagine a person traversing ...
*
Fréchet distribution The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\ \Pr(\ X \le x\ ) = e^ ~ \text ~ x > 0 ~. where is a shape para ...
*
Fréchet filter In mathematics, the Fréchet filter, also called the cofinite filter, on a set X is a certain collection of subsets of X (that is, it is a particular subset of the power set of X). A subset F of X belongs to the Fréchet filter if and only if the c ...
*
Fréchet inequalities In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George BooleBoole, G. (1854). ''An Investigation of the Laws of Thought, On Which Are Founded the Mathematical Theo ...
* Fréchet–Kuratowski theorem *
Fréchet mean In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher mean is the renaming of ...
*
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to ...
*
Fréchet–Urysohn space In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a spec ...
*
Functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
*
Topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...


Notes


External links

* * * * * {{DEFAULTSORT:Frechet, Rene Maurice Functional analysts French probability theorists French statisticians 1878 births 1973 deaths People from Yonne 20th-century French mathematicians French Protestants French mathematical analysts École Normale Supérieure alumni Lycée Buffon alumni Academic staff of the University of Strasbourg Academic staff of the University of Poitiers Fellows of the Econometric Society Members of the French Academy of Sciences Members of the Royal Netherlands Academy of Arts and Sciences French Esperantists