**Banach** founded modern functional analysis and made major contributions to the theory of topological vector spaces. In addition, he contributed to measure theory, integration, and orthogonal series.

- Stefan Greczek was born in a small village called Ostrowsko, some 50 km south of Kraków.
- Maria's guardian was a French intellectual Juliusz Mien and he quickly recognised the talents that Banach had.
- Banach attended primary school in Kraków, leaving the school in 1902 to begin his secondary education at the Henryk Sienkiewicz Gymnasium No 4 in Kraków.
- By a fortunate coincidence, one of the students in Banach's class was Witold Wilkosz who himself went on to become a professor of mathematics.
- Banach, however, remained at Henryk Sienkiewicz Gymnasium No 4 although he maintained contact with Wilkosz.
- During his first few years at the Gymnasium Banach achieved first class grades with mathematics and natural sciences being his best subjects.
- Also, while Banach was faster in mathematical problems, Wilkosz was phenomenally fast in solving problems in physics, which were of no interest to Banach.
- On leaving school Banach and Wilkosz both wanted to study mathematics, but both felt that nothing new could be discovered in mathematics so each chose to work in a subject other than mathematics.
- Banach chose to study engineering, Wilkosz chose oriental languages.
- Banach left Kraków and went to Lemberg (now Lviv in Ukraine) where he enrolled in the Faculty of Engineering at Lemberg Technical University.
- It is almost certain that Banach, without any financial support, had to support himself by tutoring.
- Lemberg was, at the time Banach studied there, under Austrian control as it had been from the partition of Poland in 1772.
- In Banach's youth Poland, in some sense, did not exist and Russia controlled much of the country.
- Banach was not physically fit for army service, having poor vision in his left eye.
- A chance event occurred in the spring of 1916 which was to have a major impact on Banach's life.
- The youngsters were Stefan Banach and Otto Nikodym.
- Steinhaus told Banach of a problem which he was working on without success.
- After a few days Banach had the main idea for the required counterexample and Steinhaus and Banach wrote a joint paper, which they presented to Zaremba for publication.
- The war delayed publication but the paper, Banach's first, appeared in the Bulletin of the Kraków Academy in 1918.
- From the time that he produced these first results with Steinhaus, Banach started to produce important mathematics papers at a rapid rate.
- Of course it is impossible to say whether, without the chance meeting with Steinhaus, Banach would have followed the route of research in mathematics.
- Banach lectured to the Society twice during 1919 and continued to produce top quality research papers.
- Banach was offered an assistantship to Lomnicki at Lwów Technical University (in what is now Lviv in Ukraine) in 1920.
- This was, of course, not the standard route to a doctorate, for Banach had no university mathematics qualifications.
- In 1922 the Jan Kazimierz University in Lwów awarded Banach his habilitation for a thesis on measure theory.
- In 1924 Banach was promoted to full professor and he spent the academic year 1924-25 in Paris.
- The years between the wars were extremely busy one for Banach.
- In 1929, together with Steinhaus, he started a new journal Studia Mathematica and Banach and Steinhaus became the first editors.
- These were set up under the editorship of Banach and Steinhaus from Lwów and Knaster, Kuratowski, Mazurkiewicz, and Sierpiński from Warsaw.
- The first volume in the series Théorie des Opérations linéaires Ⓣ(Theory of linear operations) was written by Banach and appeared in 1932.
- In 1936 Banach gave a plenary address at the International Congress of Mathematicians in Oslo.
- Another important influence on Banach was the fact that Kuratowski was appointed to the Lwów Technical University in 1927 and worked there until 1934.
- Banach collaborated with Kuratowski and they wrote some joint papers during this period.
- The way that Banach worked was unconventional.
- The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed.
- Banach had been on good terms with the Soviet mathematicians before the war started, visiting Moscow several times, and he was treated well by the new Soviet administration.
- Life at this stage was little changed for Banach who continued his research, his textbook writing, lecturing and sessions in the cafés.
- Sobolev and Aleksandrov visited Banach in Lwów in 1940, while Banach attended conferences in the Soviet Union.
- The Nazi occupation of Lwów in June 1941 meant that Banach lived under very difficult conditions.
- Towards the end of 1941 Banach worked feeding lice in German institute dealing with infectious diseases.
- As soon as the Soviet troops retook Lwów Banach renewed his contacts.
- Banach planned to go to Kraków after the war to take up the chair of mathematics at the Jagiellonian University but he died in Lwów in 1945 of lung cancer.
- Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces.
- In his dissertation, written in 1920, he defined axiomatically what today is called a Banach space.
- The name 'Banach space' was coined by Fréchet.
- Banach algebras were also named after him.
- The completeness is important as this means that Cauchy sequences in Banach spaces converge.
- Banach proved a number of fundamental results on normed linear spaces, and many important theorems are today named after him.
- There is the Hahn-Banach theorem on the extension of continuous linear functionals, the Banach-Steinhaus theorem on bounded families of mappings, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball.
- The Banach-Tarski paradox appeared in a joint paper of the two mathematicians in 1926 in Fundamenta Mathematicae entitled Sur la décomposition des ensembles de points en parties respectivement congruent Ⓣ(On the decomposition of sets of points in respectively congruent parts).
- The Banach-Tarski paradox was a major contribution to the work being done on axiomatic set theory around this period.
- Banach's open mapping of 1929 also uses set-theoretic concepts, this time concepts introduced by Baire in his 1899 dissertation.

Born 30 March 1892, Kraków, Austrian Empire (now Poland). Died 31 August 1945, Lvov, (now Lviv, Ukraine).

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Algebra, Origin Poland, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive