Stefan Banach
Encyclopedia
Stefan Banach was a Polish 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....

 who worked in interwar Poland
Second Polish Republic
The Second Polish Republic, Second Commonwealth of Poland or interwar Poland refers to Poland between the two world wars; a period in Polish history in which Poland was restored as an independent state. Officially known as the Republic of Poland or the Commonwealth of Poland , the Polish state was...

 and in Soviet Ukraine. He is generally considered to have been one of the 20th century's most important and influential mathematicians.

A self-taught mathematics prodigy
Child prodigy
A child prodigy is someone who, at an early age, masters one or more skills far beyond his or her level of maturity. One criterion for classifying prodigies is: a prodigy is a child, typically younger than 18 years old, who is performing at the level of a highly trained adult in a very demanding...

, Banach was the founder of modern 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...

 and a founder of the Lwów School of Mathematics
Lwów School of Mathematics
The Lwów School of Mathematics was a group of mathematicians who worked between the two World Wars in Lviv, then known as Lwów and located in Poland, but now located in western Ukraine. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the...

. Among his most prominent achievements was the 1932 book, Théorie des opérations linéaires (Theory of Linear Operations), the first monograph on the general theory of 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...

.

Notable mathematical concept
Concept
The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...

s named after Banach include the Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...

, the Hahn–Banach theorem
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...

, the Banach–Steinhaus theorem, the Banach-Mazur game, and the Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

.

Early life

Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in Kraków
Kraków
Kraków also Krakow, or Cracow , is the second largest and one of the oldest cities in Poland. Situated on the Vistula River in the Lesser Poland region, the city dates back to the 7th century. Kraków has traditionally been one of the leading centres of Polish academic, cultural, and artistic life...

, then part of Austro-Hungarian Galicia. Banach's parents were Stefan Greczek and Katarzyna Banach, both natives of the Podhale
Podhale
The Podhale is Poland's most southern region, sometimes referred to as the "Polish highlands". The Podhale is located in the foothills of the Tatra range of the Carpathian mountains, and is characterized by a rich tradition of folklore that is much romanticized in the Polish patriotic imagination...

 region. Stefan Greczek was born in Ostrowsko near the town of Nowy Targ
Nowy Targ
Nowy Targ is a town in southern Poland with 34,000 inhabitants , and the historical capital of the mountain region . The town is situated in the confluence of the rivers Biały and Czarny Dunajec, in a valley beneath the Gorce Mountains. It's in Nowy Targ County, Lesser Poland Voivodeship...

 and at one time was a soldier in the Austro-Hungarian Army
Austro-Hungarian Army
The Austro-Hungarian Army was the ground force of the Austro-Hungarian Dual Monarchy from 1867 to 1918. It was composed of three parts: the joint army , the Austrian Landwehr , and the Hungarian Honvédség .In the wake of fighting between the...

 stationed in Kraków. Banach's mother left him after he was baptized at four days old. Her name on the birth certificate is Katarzyna Banach.

Unusually, Stefan's surname was that of his mother instead of his father, though he received his father's given name. Since Banach's father was a private
Private (rank)
A Private is a soldier of the lowest military rank .In modern military parlance, 'Private' is shortened to 'Pte' in the United Kingdom and other Commonwealth countries and to 'Pvt.' in the United States.Notably both Sir Fitzroy MacLean and Enoch Powell are examples of, rare, rapid career...

 and was prevented by military regulations from marrying
Marriage
Marriage is a social union or legal contract between people that creates kinship. It is an institution in which interpersonal relationships, usually intimate and sexual, are acknowledged in a variety of ways, depending on the culture or subculture in which it is found...

, and the mother was too poor to support the child, the couple decided that he should be reared by family and friends.

Education

In 1902 Banach, aged 10, enrolled in Kraków's Henryk Sienkiewicz Gymnasium
Gymnasium (school)
A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...

where he became known as a prodigy
Child prodigy
A child prodigy is someone who, at an early age, masters one or more skills far beyond his or her level of maturity. One criterion for classifying prodigies is: a prodigy is a child, typically younger than 18 years old, who is performing at the level of a highly trained adult in a very demanding...

. The school specialized in the humanities. Despite this, Banach and his best friend Witold Wiłkosz, a future mathematician, would regularly work on mathematics problems during school breaks and after school. After obtaining his matura
Matura
Matura or a similar term is the common name for the high-school leaving exam or "maturity exam" in various countries, including Albania, Austria, Bosnia and Herzegovina, Bulgaria, Croatia, the Czech Republic, Hungary, Italy, Kosovo, Liechtenstein, Macedonia, Montenegro, Poland, Serbia,...

at age 18 in 1910, Banach went with Witold Wiłkosz to Lwów, then the capital of Galicia, with the intention of studying engineering at the Lwów Polytechnic. However, as Banach had to earn money to support his studies, it was not until 1914 that he finally, at age 22, passed his high school graduation exams.

When World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...

 broke out, Banach was excused from military service due to his left-handedness and poor vision. When the Russian Army opened its offensive toward Lwów, Banach left for Kraków, to spend the rest of the war there. He made his living tutor
Tutor
A tutor is a person employed in the education of others, either individually or in groups. To tutor is to perform the functions of a tutor.-Teaching assistance:...

ing at local gymnasium
Gymnasium (school)
A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...

s and working in a bookshop. He may have attended lectures at the Jagiellonian University
Jagiellonian University
The Jagiellonian University was established in 1364 by Casimir III the Great in Kazimierz . It is the oldest university in Poland, the second oldest university in Central Europe and one of the oldest universities in the world....

, but little is known of that period in his life.

In 1916, in Kraków's Planty
Planty Park
Planty is one of the largest city parks in Kraków, Poland. It encircles the Stare Miasto , where the Medieval city walls used to stand until the early 19th century. The historic Old Town is not to be confused with the Administrative District No. 1 Stare Miasto extending further east.The park has an...

gardens, Banach encountered Professor Hugo Steinhaus
Hugo Steinhaus
Władysław Hugo Dionizy Steinhaus was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics...

, one of the renowned mathematicians of the time. Steinhaus became fascinated with the self-taught young mathematician. The encounter resulted in a long-lasting collaboration and friendship. It was also through Steinhaus that Banach met his future wife, Łucja Braus.

Work

Steinhaus introduced Banach to academic circles and substantially accelerated his career. After Poland regained independence, in 1920 Banach was given an assistantship at Kraków's Jagiellonian University
Jagiellonian University
The Jagiellonian University was established in 1364 by Casimir III the Great in Kazimierz . It is the oldest university in Poland, the second oldest university in Central Europe and one of the oldest universities in the world....

. Steinhaus' backing also allowed him to receive a doctorate
Doctorate
A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder to teach in a specific field, A doctorate is an academic degree or professional degree that in most countries refers to a class of degrees which qualify the holder...

 without actually graduating from a university. The doctoral thesis, accepted by King John II Casimir University of Lwów
Lviv University
The Lviv University or officially the Ivan Franko National University of Lviv is the oldest continuously operating university in Ukraine...

 and published in 1922, included the basic ideas of 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...

, which was soon to become an entirely new branch of mathematics. The thesis was widely discussed in academic circles and allowed him in 1922 to become a professor at the Lwów Polytechnic. Initially an assistant to Professor Antoni Łomnicki, in 1927 Banach received his own chair. In 1924 he was also accepted as a member of the Polish Academy of Learning
Polish Academy of Learning
The Polish Academy of Arts and Sciences or Polish Academy of Learning , headquartered in Kraków, is one of two institutions in contemporary Poland having the nature of an academy of sciences....

. At the same time, from 1922, Banach also headed the second Chair of Mathematics at University of Lwów
Lviv University
The Lviv University or officially the Ivan Franko National University of Lviv is the oldest continuously operating university in Ukraine...

.

Young and talented, Banach gathered around him a large group of mathematicians. The group, meeting in the Scottish Café
Scottish Café
The Scottish Café was the café in Lwów where, in the 1930s and 1940s, mathematicians from the Lwów School collaboratively discussed research problems, particularly in functional analysis and topology....

, soon gave birth to the "Lwów School of Mathematics
Lwów School of Mathematics
The Lwów School of Mathematics was a group of mathematicians who worked between the two World Wars in Lviv, then known as Lwów and located in Poland, but now located in western Ukraine. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the...

". In 1929 the group began publishing its own journal, Studia Mathematica
Studia Mathematica
Studia Mathematica is a Polish mathematics journal published since 1929. It was founded by Stefan Banach and Hugo Steinhaus. It accepts papers written in English, French, German, or Russian, primarily in functional analysis and related areas such as probability theory.- External links :* *...

, devoted primarily to Banach's field of study — functional analysis. Around that time, Banach also began working on his best-known work, the first monograph on the general theory of linear-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...

. First published in Polish in 1931, the following year it was also translated into French and gained wider recognition in European academic circles. The book was also the first in a long series of mathematics monographs edited by Banach and his circle.

Later life and death

Following the invasion of Poland
Invasion of Poland (1939)
The Invasion of Poland, also known as the September Campaign or 1939 Defensive War in Poland and the Poland Campaign in Germany, was an invasion of Poland by Germany, the Soviet Union, and a small Slovak contingent that marked the start of World War II in Europe...

 by Nazi Germany
Nazi Germany
Nazi Germany , also known as the Third Reich , but officially called German Reich from 1933 to 1943 and Greater German Reich from 26 June 1943 onward, is the name commonly used to refer to the state of Germany from 1933 to 1945, when it was a totalitarian dictatorship ruled by...

 and the Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....

, Lwów came under the control of the Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....

 for almost two years. Banach, from 1939 a corresponding member of the Academy of Sciences of Ukraine, and on good terms with Soviet
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....

 mathematicians, had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities. Following the German takeover of Lwów in 1941 in Operation Barbarossa
Operation Barbarossa
Operation Barbarossa was the code name for Germany's invasion of the Soviet Union during World War II that began on 22 June 1941. Over 4.5 million troops of the Axis powers invaded the USSR along a front., the largest invasion in the history of warfare...

, all universities were closed and Banach, along with many colleagues and his son, was employed as lice feeder at Professor Rudolf Weigl
Rudolf Weigl
Professor Rudolf Stefan Weigl was a famous Polish biologist and inventor of the first effective vaccine against epidemic typhus. Weigl founded the Weigl Institute in Lwów, Poland , where he did his vaccine-producing research.Of Austrian ethnic descent, Weigl was born in Přerov, Moravia...

's Typhus
Typhus
Epidemic typhus is a form of typhus so named because the disease often causes epidemics following wars and natural disasters...

 Research Institute. Employment in Weigl's Institute provided many unemployed university professors and their associates protection from random arrest and deportation to Nazi concentration camps
Nazi concentration camps
Nazi Germany maintained concentration camps throughout the territories it controlled. The first Nazi concentration camps set up in Germany were greatly expanded after the Reichstag fire of 1933, and were intended to hold political prisoners and opponents of the regime...

.

After the Red Army
Red Army
The Workers' and Peasants' Red Army started out as the Soviet Union's revolutionary communist combat groups during the Russian Civil War of 1918-1922. It grew into the national army of the Soviet Union. By the 1930s the Red Army was among the largest armies in history.The "Red Army" name refers to...

 recaptured Lviv in the Lvov–Sandomierz Offensive of 1944, Banach returned to the University and helped re-establish it after the war years. However, because the Soviets were removing Poles
Repatriation of Poles (1944–1946)
The Polish population transfers from the former eastern territories of Poland also known as the flight and expulsion of Poles towards the end – and in the aftermath – of World War II refer to the forced migration of Poles between 1944–1946...

 from annexed formerly Polish territories, Banach began preparing to leave the city and settle in Kraków
Kraków
Kraków also Krakow, or Cracow , is the second largest and one of the oldest cities in Poland. Situated on the Vistula River in the Lesser Poland region, the city dates back to the 7th century. Kraków has traditionally been one of the leading centres of Polish academic, cultural, and artistic life...

, Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

, where he had been promised a chair at the Jagiellonian University
Jagiellonian University
The Jagiellonian University was established in 1364 by Casimir III the Great in Kazimierz . It is the oldest university in Poland, the second oldest university in Central Europe and one of the oldest universities in the world....

. He was also considered a candidate for Minister of Education of Poland. In January 1945, however, he was diagnosed with lung cancer and was allowed to stay in Lwów. He died on August 31, 1945, aged 53. His funeral at the Lychakiv Cemetery turned into a patriotic demonstration by the Poles who still remained in the city.

Works

  • Rachunek różniczkowy i całkowy, tom I (Differential and Integral Calculus, vol. 1), Lwów, Zakład Narodowy im. Ossolińskich, 1929, 294 pp.
  • Rachunek różniczkowy i całkowy, tom II (Differential and Integral Calculus, vol. 2), Lwów, Książnica-Atlas, 1930, 248 pp.
  • Teoria operacji. Tom l. Operacje liniowe (Theory of operations, vol. 1: Linear operations), Warsaw, Kasa im. Mianowskiego, 1931, viii + 236 pp.
  • Théorie des opérations linéaires, Monografie Matematyczne 1 (Theory of Linear Operations, Mathematical Monographs 1), Warsaw, 1932, vii + 254 pp.
  • Mechanika w zakresie szkół akademickich, Monografie Matematyczne 8 (Mechanics for Academic Schools, Mathematical Monographs 8), Warsaw, Lwów, Wilno, 1938.


Banach's most influential work was Théorie des opérations linéaires (Theory of Linear Operations, 1932). In it he formulated the concept now known as "Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

", and proved many fundamental theorems of 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...

.

Besides being one of the founders of functional analysis, Banach also made important contributions to measure theory, set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, and other branches of mathematics.

He was also one of the founders and editors of the journal, Studia Mathematica
Studia Mathematica
Studia Mathematica is a Polish mathematics journal published since 1929. It was founded by Stefan Banach and Hugo Steinhaus. It accepts papers written in English, French, German, or Russian, primarily in functional analysis and related areas such as probability theory.- External links :* *...

.

Quotes

Stanisław Ulam, another mathematician of the Lwów School of Mathematics
Lwów School of Mathematics
The Lwów School of Mathematics was a group of mathematicians who worked between the two World Wars in Lviv, then known as Lwów and located in Poland, but now located in western Ukraine. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the...

, in his autobiography, quotes Banach as saying:
"Good mathematicians see analogies. Great mathematicians see analogies between analogies."


Hugo Steinhaus
Hugo Steinhaus
Władysław Hugo Dionizy Steinhaus was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics...

 said of Banach:
"An exceptional intellect, exceptional discoveries... he gave Polish science... more than anybody else."
"Banach was my greatest scientific discovery."

See also

  • 16856 Banach
    16856 Banach
    16856 Banach is a main-belt asteroid discovered on December 28, 1997 by P. G. Comba at Prescott.It is named after Polish mathematician Stefan Banach.- External links :*...

  • Amenable Banach algebra
    Amenable Banach algebra
    A Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner ....

  • Banach–Alaoglu theorem
  • Banach algebra
    Banach algebra
    In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...

  • Banach bundle
    Banach bundle
    In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.-Definition of a Banach bundle:...

  • Banach fixed point theorem
    Banach fixed point theorem
    In mathematics, the Banach fixed-point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points...

  • Banach function algebra
    Banach function algebra
    In functional analysis a Banach function algebra on a compact Hausdorff space X is unital subalgebra, A of the commutative C*-algebra C of all continuous, complex valued functions from X, together with a norm on A which makes it a Banach algebra.For every x\in X define \varepsilon_x=f\...

  • Banach integral
  • Banach limit
  • Banach manifold
    Banach manifold
    In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space...

  • Banach's matchbox problem
    Banach's matchbox problem
    Banach's match problem is a classic problem in probability attributed to Stefan Banach. Feller says that the problem was inspired by a humorous reference to Banach's smoking habit in a speech honouring him by H...

  • Banach measure
    Banach measure
    In mathematics, Banach measure in measure theory may mean a real-valued function on the algebra of all sets , by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area. That is, this is a theoretical definition getting round...

  • Banach space
    Banach space
    In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

  • Banach–Mazur game
    Banach–Mazur game
    In general topology, set theory and game theory, a Banach–Mazur game is a topological game played by two players, trying to pin down elements in a set . The concept of a Banach–Mazur game is closely related to the concept of Baire spaces...

  • Banach–Mazur theorem
  • Banach–Schauder theorem
  • Banach–Stone theorem
  • Banach–Tarski paradox
    Banach–Tarski paradox
    The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...

  • Closed range theorem
    Closed range theorem
    In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range...

  • 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...

  • Legitimacy
  • List of Poles

External links

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