René-Louis Baire
Encyclopedia
René-Louis Baire was a French
mathematician
most famous for his Baire category theorem
, which helped to generalize and prove future theorems. His theory was published originally in his dissertation Sur les fonctions de variable réelles ("On the Functions of Real Variables") in 1899.
through the use of a scholarship. In 1890, Baire completed his advanced classes and entered the special mathematics section of the Lycée Henri IV
. While there, he prepared for and passed the entrance examination for the École Normale Supérieure
and the École Polytechnique
. He decided to attend the École Normale Supérieure in 1891. After receiving his three year degree, Baire proceeded toward his agregation
. He did better than all the other students on the writing portion of the test but he did not pass the oral examination due to a lack of explanation and clarity in his lesson. After retaking the agregation and passing, he was assigned to teach at the Lycée in Bar-le-Duc. While there, Baire researched the concept of limits and discontinuity for his doctorate. He presented his thesis on March 24, 1899 and was awarded his doctorate. He continued to teach in lycées around France but was not happy teaching lower level mathematics. In 1901 Baire was appointed to the University of Montpellier
as a "Maître de conférences". In 1904 he was awarded a Peccot Foundation Fellowship to spend a semester in a university and develop his skills as a professor. Baire chose to attend the Collège de France
where he lectured on the subject of analysis. He was appointed to a university post in 1905 when he joined the Faculty of Science at Dijon. In 1907 he was promoted to Professor of Analysis at Dijon where he continued his research in analysis. He retired from Dijon in 1925 and spent his last years living in multiple hotels that he could afford with his meager pension.
. From time to time, his health would prevent him from working or studying. The bad spells became more frequent, immobilizing him for long periods of time. Over time, he had developed a kind of psychological disorder which made him unable to undertake work which required long periods of concentration. At times this would make his ability to research mathematics impossible. Between 1909 and 1914 this problem continually plagued him and his teaching duties became more and more difficult. He was given a leave of absence from Dijon due to all these breakdowns.
and Lebesgue. In his dissertation Sur les fonctions de variable réelles ("On the Functions of Real Variables"), Baire studied a combination of set theory and analysis topics to arrive at the Baire Category Theorem
and define the nowhere dense set
. He then used these topics to prove the theorems of those he studied with and further the understanding of continuity. Among Baire’s other most important works are Théorie des nombres irrationels, des limites et de la continuité (Theory of Irrational Numbers, Limits, and Continuity) published in 1905 and both volumes of Leçons sur les théories générales de l’analyse (Lessons on the General Theory of Analysis) published in 1907–08.
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....
most famous for his Baire category theorem
Baire category theorem
The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
, which helped to generalize and prove future theorems. His theory was published originally in his dissertation Sur les fonctions de variable réelles ("On the Functions of Real Variables") in 1899.
Schooling
The son of a tailor, Baire was one of three children from a poor working class family in Paris. He started his studies when he entered the Lycée LakanalLycée Lakanal
Lycée Lakanal is a secondary public school in Sceaux, France. It was named after Joseph Lakanal, a French politician, and an original member of the Institut de France. The school also offers a middle school and highly ranked "classes préparatoires" undergraduate training...
through the use of a scholarship. In 1890, Baire completed his advanced classes and entered the special mathematics section of the Lycée Henri IV
Lycée Henri IV
The Lycée Henri-IV is a public secondary school located in Paris. Along with Louis-le-Grand it is widely regarded as one of the most demanding sixth-form colleges in France....
. While there, he prepared for and passed the entrance examination for the École Normale Supérieure
École Normale Supérieure
The École normale supérieure is one of the most prestigious French grandes écoles...
and the École Polytechnique
École Polytechnique
The École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...
. He decided to attend the École Normale Supérieure in 1891. After receiving his three year degree, Baire proceeded toward his agregation
Agrégation
In France, the agrégation is a civil service competitive examination for some positions in the public education system. The laureates are known as agrégés...
. He did better than all the other students on the writing portion of the test but he did not pass the oral examination due to a lack of explanation and clarity in his lesson. After retaking the agregation and passing, he was assigned to teach at the Lycée in Bar-le-Duc. While there, Baire researched the concept of limits and discontinuity for his doctorate. He presented his thesis on March 24, 1899 and was awarded his doctorate. He continued to teach in lycées around France but was not happy teaching lower level mathematics. In 1901 Baire was appointed to the University of Montpellier
University of Montpellier
The University of Montpellier was a French university in Montpellier in the Languedoc-Roussillon région of the south of France. Its present-day successor universities are the University of Montpellier 1, Montpellier 2 University and Paul Valéry University, Montpellier III.-History:The university...
as a "Maître de conférences". In 1904 he was awarded a Peccot Foundation Fellowship to spend a semester in a university and develop his skills as a professor. Baire chose to attend the Collège de France
Collège de France
The Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue Saint-Jacques and Rue des Écoles...
where he lectured on the subject of analysis. He was appointed to a university post in 1905 when he joined the Faculty of Science at Dijon. In 1907 he was promoted to Professor of Analysis at Dijon where he continued his research in analysis. He retired from Dijon in 1925 and spent his last years living in multiple hotels that he could afford with his meager pension.
Illness
Since he was young, Baire always had "delicate" health. He had developed problems with his esophagus before he attended school and he would occasionally experience severe attacks of agoraphobiaAgoraphobia
Agoraphobia is an anxiety disorder defined as a morbid fear of having a panic attack or panic-like symptoms in a situation from which it is perceived to be difficult to escape. These situations can include, but are not limited to, wide-open spaces, crowds, or uncontrolled social conditions...
. From time to time, his health would prevent him from working or studying. The bad spells became more frequent, immobilizing him for long periods of time. Over time, he had developed a kind of psychological disorder which made him unable to undertake work which required long periods of concentration. At times this would make his ability to research mathematics impossible. Between 1909 and 1914 this problem continually plagued him and his teaching duties became more and more difficult. He was given a leave of absence from Dijon due to all these breakdowns.
Contributions to Mathematics
Baire's skill in mathematical analysis led him to study with other major names in analysis such as Vito VolterraVito Volterra
Vito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....
and Lebesgue. In his dissertation Sur les fonctions de variable réelles ("On the Functions of Real Variables"), Baire studied a combination of set theory and analysis topics to arrive at the Baire Category Theorem
Baire category theorem
The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
and define the nowhere dense set
Nowhere dense set
In mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it...
. He then used these topics to prove the theorems of those he studied with and further the understanding of continuity. Among Baire’s other most important works are Théorie des nombres irrationels, des limites et de la continuité (Theory of Irrational Numbers, Limits, and Continuity) published in 1905 and both volumes of Leçons sur les théories générales de l’analyse (Lessons on the General Theory of Analysis) published in 1907–08.
See also
- Baire functionBaire functionIn mathematics, Baire functions are certain sets of functions. They are studied in several fields of mathematics, including real analysis andtopology....
- Baire measureBaire measureA Baire measure is a measure on the σ-algebra of Baire sets of a topological space. In spaces that are not metric spaces, the Borel sets and the Baire sets may differ...
- Baire setBaire setIn mathematics, more specifically in measure theory, the notion of a Baire set is important in the understanding of particular relations between measure theory and topology. In particular, an understanding of Baire sets aids in intuition when one deals with measures on non-metrizable topological...
- Baire spaceBaire spaceIn mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
- Baire space (set theory)Baire space (set theory)In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called “reals.” It is often denoted B, N'N, or ωω...
- Property of Baire
- normal convergenceNormal convergenceIn mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.- History :...
- Nowhere dense setNowhere dense setIn mathematics, a nowhere dense set in a topological space is a set whose closure has empty interior. The order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it...
- Baire Category TheoremBaire category theoremThe Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....