# Stefan Banach

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**Stefan Banach** ( [,stɛfan 'banax] (help·info); 1892-1945) was an eminent Polish mathematician and university professor. A self-taught mathematical prodigy, Banach was a founder of functional analysis and of the Lwów School of Mathematics. Among his most prominent achievements was a 1932 book, *Théorie des opérations linéaires* (Theory of Linear Operations), the first monograph on the general theory of linear-metric space.

Notable mathematical concepts named after Banach include the Banach–Tarski paradox, Hahn–Banach theorem, Banach-Steinhaus theorem, and Banach space.

## Life

Stefan Banach was born March 30, 1892, in Kraków, then part of Austro-Hungarian Galicia. Banach's parents were Stefan Greczek, a young soldier in the Austro-Hungarian Army, and Katarzyna Banach, both natives of the Podhale region.^{[1]}. Since Banach's father was a private and was prevented by military regulations from marrying, and the child's mother was too poor to support young Stefan, the couple decided that the child should be reared by a friend of Banach's father, the owner of a Kraków laundry.^{[1]} Stefan Greczek paid for his son's education and would be the only relative whom Banach would personally know.^{[1]}

Already as a student at Kraków's *Gymnasium* no. IV, Banach became known as a prodigy. In 1906, aged 14, he was studying higher mathematics, and two years later he had started in on several languages, western and eastern.^{[2]} After obtaining his *matura* at age 18 (1910), Banach went with his friend Witold Wiłkosz to Lwów (now Lviv, Ukraine; then the capital of Galicia), intending to enroll in 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 half-diploma exams.^{[3]}

When World War I 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 and in other Galician towns. He made his living tutoring at local *gymnasium*s and working in a bookshop. He may have attended lectures at Kraków University, but little is known of that period in his life.^{[4]}

In 1916, in Kraków's *Planty* gardens, Banach encountered Professor Hugo Steinhaus, one of the most renowned mathematicians of the age. 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.

Steinhaus introduced Banach to academic circles and significantly accelerated his career. After Poland regained independence, in 1920 Banach was given assistantship at the Jagiellonian University. Steinhaus' protection also allowed him to receive a doctorate without actually graduating from any university. The doctorate's thesis, accepted by the Jan Kazimierz University of Lwów and published in 1922,^{[5]} included the basic ideas of the functional analysis, that was soon to become an entirely new branch of mathematics. The thesis was widely discussed in academic circles and allowed him to become a professor at the Lwów University of Science and Technology in 1922. Initially an assistant to Prof. Antoni Łomnicki, in 1927 Banach received his own chair. In 1924 he was also accepted as a member of the Polish Academy of Skills. At the same time, since 1922, Banach also headed the second Chair of Mathematics at the Jan Kazimierz University.

Young and talented, Banach gathered around him a large group of mathematicians. The group, meeting in the Scottish Café, soon gave birth to the "Lwów School of Mathematics." In 1929 the group began publishing its own journal, *Studia Mathematica*, 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. First published in Polish in 1931,^{[6]} the following year it was also translated into French and gained wider recognition in European academic circles.^{[7]} The book was also the first in a long series of mathematics monographs edited by Banach and his circle.

Following the Invasion of Poland (1939) by Nazi Germany and the Soviet Union, Lwów was occupied by the latter state. Banach, being a corresponding member of the Academy of Sciences of the Ukrainian SSR, and otherwise on good terms with Soviet mathematicians, was allowed to keep his chair and continue his academic activities. Following the German takeover of Lwów in 1941, all universities were closed and Banach, along with many colleagues and his son, was forced to eke out a living feeding lice with his blood at Professor Rudolf Weigl's Typhus Research Institute.

After the second Soviet occupation of Lwów commenced in 1944, Banach returned to the university and helped re-establish it after the war years. However, as it was decided to expel all PolesTemplate:Fix/category^{[citation needed]} from the territories that were already annexed by the Soviets, Banach began preparing to leave the city and settle in Kraków,Template:Fix/category^{[citation needed]} where he had been promised a chair at the Jagiellonian University. However, in January 1945 he was diagnosed with lung cancer and was allowed to stay in Lwów. He died August 31, 1945, aged 53. His funeral at Lwów's Łyczaków Cemetery turned into a patriotic demonstration by the Poles who still remained in the city.

## Work

*Théorie des opérations linéaires* (*Teoria operacji liniowych*, 1932) is regarded as Banach's most influential work. In it he formulated the concept now known as Banach space, and proved many fundamental theorems of functional analysis. He was also one of the founders and editors the journal *Studia Mathematica*.

Besides being one of the founders of functional analysis, Banach also made important contributions to measure theory, set theory, and other branches of mathematics.

## Quotes

Stanisław Marcin Ulam, another mathematician of the Lwów School of Mathematics, in his autobiography, attributes this to Banach:

- "Good mathematicians see analogies. Great mathematicians see analogies between analogies."

Hugo Steinhaus said of Banach:

- "An exceptional intellect, exceptional discoveries...he gave Polish science...more than anybody else."

- "Banach was my greatest scientific discovery."

**Stefan Banach**:

- "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

## Notes and references

**In-line:**

- ↑
^{1.0}^{1.1}^{1.2}**Invalid language code.**Monika Waksmundzka-Hajnos (2006). "Wspomnienie o Stefanie Greczku".*Wortal Stefana Banacha*. Gdańsk University. http://banach.univ.gda.pl/greczek.html#start. Retrieved 2007-02-22. - ↑
**Invalid language code.**Roman Kałuża (1996).*Through a Reporter's Eyes: the life of Stefan Banach*. pp. 137. ISBN 0817643710. http://books.google.com/books?vid=ISBN0817637729&id=i3ZrfMinChkC&printsec=frontcover&dq=Stefan+Banach#PPP1,M1. - ↑ Roman Kałuża, op.cit., p.13
- ↑ Roman Kałuża, op.cit., p.16
- ↑
**Invalid language code.****Invalid language code.**Stefan Banach (1922). "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales".*Fundamenta Mathematicae***III**. - ↑ Stefan Banach:
*Teoria operacji liniowych*. - ↑
**Invalid language code.**Stefan Banach:*Théorie des opérations linéaires*(Theory of Linear Operations).

## See also

- Banach algebra
- Hahn-Banach theorem
- Banach fixed point theorem
- Banach-Schauder theorem
- Banach-Alaoglu theorem

## External links

- Page devoted to Stefan Banach
- Théorie des opérations linéaires French translation from 1932
- The MacTutor History of Mathematics archive Article on Stefan Banach
- Stefan Banach at the Mathematics Genealogy Project