Stefan Banach

BanachBanach, StefanBanachaS. Banach
Stefan Banach (Polish: ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.wikipedia
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Hugo Steinhaus

SteinhausHugo Dyonizy Steinhaussection below
However, during World War I Banach returned to Kraków, where he befriended Hugo Steinhaus.
He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach–Steinhaus theorem.

Functional analysis

functionalfunctional analyticalgebraic function theory
He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics.
Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.

Poles

PolishPolePolish people
Stefan Banach (Polish: ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.
Outstanding Polish mathematicians formed the Lwów School of Mathematics (including Stefan Banach, Hugo Steinhaus, Stanisław Ulam) and Warsaw School of Mathematics (including Alfred Tarski, Kazimierz Kuratowski, Wacław Sierpiński).

Witold Wilkosz

Wilkosz, Witold
Born in Kraków, Banach attended IV Gymnasium, a secondary school, and worked on mathematics problems with his friend Witold Wilkosz.
He was a friend of fellow mathematician Stefan Banach.

Feeder of lice

lice feederfeeders of lice
As a result, Banach was forced to earn a living as a feeder of lice at Rudolf Weigl's Institute for Study of Typhus and Virology.
Initially begun in 1920 by Weigl, during the German occupation of the city it became the primary means of support and protection for many of the city's Polish intellectuals, including the mathematician Stefan Banach and the poet Zbigniew Herbert.

Banach algebra

Banach algebrasspectral mapping theoremalgebra norm
Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.

Banach space

Banach spacesBanachBanach-space theory
Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.
Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.

Lwów School of Mathematics

groupLwow School of Mathematicsmathematicians from the Lwów School
He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics.

Banach–Tarski paradox

Banach-Tarski paradoxBanach-TarskiBanach-Tarski decomposition
Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.
In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions.

Hahn–Banach theorem

Hahn-Banach theoremHahn–Banach separation theorem
Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.
The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s.

Banach–Mazur game

Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.
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 (space).

Banach fixed-point theorem

Banach fixed point theoremBanach contraction principleContraction mapping theorem
Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem. Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students (for example in the Banach–Schauder theorem) and other mathematicians (in particular Brouwer and Poincaré and Birkhoff).
The theorem is named after Stefan Banach (1892–1945) and Renato Caccioppoli (1904–1959), and was first stated by Banach in 1922.

Banach–Alaoglu theorem

Alaoglu's theoremBourbaki–Alaoglu theorem
Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.
A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach, and the first proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.

Kazimierz Żorawski

Kazimierz ŻórawskiŻorawski, Kazimierz
He attended some lectures at the Jagiellonian University at that time, including those of the famous Polish mathematicians Stanisław Zaremba and Kazimierz Żorawski, but little is known of that period of his life.
His work earned him an honored place in mathematics alongside such Polish mathematicians as Wojciech Brudzewski, Jan Brożek (Broscius), Nicolas Copernicus, Samuel Dickstein, Stefan Banach, Stefan Bergman, Marian Rejewski, Wacław Sierpiński, Stanisław Zaremba and Witold Hurewicz.

Banach measure

finitely additive measure
Some of the notable mathematical concepts that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, and the Banach fixed-point theorem.
Stefan Banach showed that it is possible to define a Banach measure for the Euclidean plane, consistent with the usual Lebesgue measure.

Vector space

vectorvector spacesvectors
Around that time, Banach also began working on his best-known work, the first monograph on the general theory of linear-metric space.
This was later formalized by Banach and Hilbert, around 1920.

Scottish Café

Scottish bookScottish cafe
The group, meeting in the Scottish Café, soon gave birth to the "Lwów School of Mathematics".
To keep the results from being lost, and after becoming annoyed with their writing directly on the table tops, Stefan Banach's wife provided the mathematicians with a large notebook, which was used for writing the problems and answers and eventually became known as the Scottish Book.

Open mapping theorem (functional analysis)

open mapping theoremBanach–Schauder theorem
Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students (for example in the Banach–Schauder theorem) and other mathematicians (in particular Brouwer and Poincaré and Birkhoff).
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.

Włodzimierz Stożek

Stożek, WłodzimierzWlodzimierz Stozek
Steinhaus, Banach and Nikodym, along with several other Kraków mathematicians (Władysław Ślebodziński, Leon Chwistek, Alfred Rosenblatt and Włodzimierz Stożek) also established a mathematical society, which eventually became the Polish Mathematical Society.
In December 1944, Stefan Banach wrote the following tribute to Stożek:

Studia Mathematica

Studia Math.
In 1929 the group began publishing its own journal, Studia Mathematica, devoted primarily to Banach's field of study—functional analysis.
The journal was established in 1929 by Stefan Banach and Hugo Steinhaus and its first editors were Banach, Steinhaus and Herman Auerbach.

List of things named after Stefan Banach

Stefan Banach was a Polish mathematician who made key contributions to mathematics.

Gorals

GóralGoralGórale
Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in Kraków, then part of the Austro-Hungarian Empire, into a Góral Roman Catholic family and was subsequently baptised by his father, while his mother abandoned him upon this event and her identity is ambiguous.

Antoni Łomnicki

Antoni LomnickiŁomnicki, Antoni
Initially an assistant to Professor Antoni Łomnicki, in 1927 Banach received his own chair.
In December 1944 Stefan Banach wrote the following tribute to Łomnicki:

Banach–Stone theorem

Banach-Stone theorem
In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.