Maurice René Fréchet

# Maurice René Fréchet

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Maurice Fréchet (September 2, 1878 – June 4, 1973) was a French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He made major contributions to the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

of point sets and introduced the entire concept of metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

s. He also made several important contributions to the field of statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

and probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, as well as calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. His dissertation opened the entire field of functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...

s on metric spaces and introduced the notion of compactness. Independently of Riesz
Riesz
Riesz may refer to:* Frigyes Riesz, mathematician* Marcel Riesz, mathematician, younger brother of Frigyes Riesz* Riesz, a fictional character in the video game Seiken Densetsu 3- See also :...

, he discovered the representation theorem in the space of Lebesgue square integrable functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

.

### Early life

He was born to a Protestant family in Maligny, Yonne
Maligny, Yonne
Maligny is a commune in the Yonne department in Burgundy in north-central France....

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
French Third Republic
The French Third Republic was the republican government of France from 1870, when the Second French Empire collapsed due to the French defeat in the Franco-Prussian War, to 1940, when France was overrun by Nazi Germany during World War II, resulting in the German and Italian occupations of France...

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
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

. 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
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. Jean-Claude Durand is its current proviseur....

in Paris where he was taught mathematics by Jacques Hadamard
Jacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...

. Hadamard recognised the potential of young Maurice and decided to tutor him on an individual basis. After Hadamard moved to the University of Bordeaux
University of Bordeaux
University of Bordeaux is an association of higher education institutions in and around Bordeaux, France. Its current incarnation was established 21 March 2007. The group is the largest system of higher education schools in southwestern France. It is part of the Academy of Bordeaux.There are seven...

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
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...

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
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...

due to his contact with American mathematicians in Paris—particularly Edwin Wilson
Edwin Bidwell Wilson
Edwin Bidwell Wilson was an American mathematician and polymath. He was the sole protégé of Yale's physicist Josiah Willard Gibbs and was mentor to MIT economist Paul Samuelson. He received his AB from Harvard College in 1899 and his PhD from Yale University in 1901, working under Gibbs.E.B...

.

### 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
Besançon
Besançon , is the capital and principal city of the Franche-Comté region in eastern France. It had a population of about 237,000 inhabitants in the metropolitan area in 2008...

, then moved in 1908 to the Lycée in Nantes
Nantes
Nantes is a city in western France, located on the Loire River, from the Atlantic coast. The city is the 6th largest in France, while its metropolitan area ranks 8th with over 800,000 inhabitants....

to stay there for a year. After that he served at the University of Poitiers
Poitiers
Poitiers is a city on the Clain river in west central France. It is a commune and the capital of the Vienne department and of the Poitou-Charentes region. The centre is picturesque and its streets are interesting for predominant remains of historical architecture, especially from the Romanesque...

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
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...

at the University of Illinois
University of Illinois at Urbana-Champaign
The University of Illinois at Urbana–Champaign is a large public research-intensive university in the state of Illinois, United States. It is the flagship campus of the University of Illinois system...

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
British Army
The British Army is the land warfare branch of Her Majesty's Armed Forces in the United Kingdom. It came into being with the unification of the Kingdom of England and Scotland into the Kingdom of Great Britain in 1707. The new British Army incorporated Regiments that had already existed in England...

. 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
Strasbourg
Strasbourg is the capital and principal city of the Alsace region in eastern France and is the official seat of the European Parliament. Located close to the border with Germany, it is the capital of the Bas-Rhin département. The city and the region of Alsace are historically German-speaking,...

to help with the reestablishment of the university
University of Strasbourg
The University of Strasbourg in Strasbourg, Alsace, France, is the largest university in France, with about 43,000 students and over 4,000 researchers....

. 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
Émile Borel
Félix Édouard Justin Émile Borel was a French mathematician and politician.Borel was born in Saint-Affrique, Aveyron. Along with René-Louis Baire and Henri Lebesgue, he was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his...

, 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
An Academy of Sciences is a national academy or another learned society dedicated to sciences.In non-English speaking countries, the range of academic fields of the members of a national Academy of Science often includes fields which would not normally be classed as "science" in English...

until the age of 78.

Fréchet was an Esperantist
Esperantist
An Esperantist is a person who speaks or uses Esperanto. Etymologically, an Esperantist is someone who hopes...

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

## Main works

His first major work was his outstanding PhD
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...

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
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

, although the name is due to Hausdorff
Felix Hausdorff
Felix Hausdorff was a Jewish German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.-Life:Hausdorff studied at the University of Leipzig,...

. The level of abstraction used by Fréchet is similar to that in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, 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 developed here ideas from the article Deux types fondamentaux de distribution statistique (1938; an English translation Two Basic Types of Statistical Distribution) of Czech geographer
Geographer
A geographer is a scholar whose area of study is geography, the study of Earth's natural environment and human society.Although geographers are historically known as people who make maps, map making is actually the field of study of cartography, a subset of geography...

, demographer and statistician
Statistician
A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...

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 (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 space
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...

• Fréchet–Urysohn space
• Fréchet derivative
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...

• Differentiation in Fréchet spaces
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 is significantly weaker than the derivative in a Banach space. Nevertheless, it is the weakest notion of...

• Topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

• Functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

• Fréchet distribution
• Fréchet mean
Fréchet mean
The Fréchet mean , is the point, x, that minimizes the Fréchet function, in cases where such a unique minimizer exists. The value at a point p, of the Fréchet function associated to a random point X on a complete metric space is the expected squared distance from p to X...

• Fréchet distance
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...

• Fréchet filter
Fréchet filter
In mathematics, Fréchet filter is an important concept in order theory.If X is an infinite set, the Fréchet filter,also called cofinite filter on X, is the set of all subsets of X having finite complement in X....