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Stefan Banach (1892‚Äď1945) was a Polish mathematician who worked in interwar Polandmarker and in Soviet Ukraine.

A self-taught mathematics prodigy, Banach was the founder of modern functional analysis and a founder of the Lwów School of Mathematics. 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 linear-metric space.

Notable mathematical concepts named after Banach include the Banach‚ÄďTarski paradox, Hahn‚ÄďBanach theorem, Banach‚ÄďSteinhaus theorem, Banach-Mazur game and Banach space.


Stefan Banach was born on March 30, 1892, at St. Lazarus General Hospital in Krakówmarker, then part of Austro-Hungarian Galicia. Banach's parents were Stefan Greczek and one Katarzyna Banach, both natives of the Podhale region. Stefan Greczek was born in Ostrowskomarker near the town of Nowy Targmarker and at one time was a soldier in the Austro-Hungarian Army stationed in Kraków. Stefan Greczek's father, Józef, was a farmer and a village mayor and Józef's wife, Antonina (née Borkowska) bore the Pomian coat of arms.

Banach's mother left him after baptizing him when he was four days old. Her name on the birth certificate is Katarzyna Banach. Later in life Banach would ask his father to tell him his mother's actual identity but would only be told that he had taken an oath of secrecy about it. Stefan Greczek would go on to marry twice and have a son by his first wife and four children by the second.

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 and was prevented by military regulations from marrying, and the mother was too poor to support the child, the couple decided that he should be reared by family and friends. Family legend says that Banach spent his early childhood in Ostrowsko with his grandmother, to whom he was very close. When she became ill, his father sent him to Krak√≥w to live with Franciszka PŇāowa and her daughter, Maria, although Banach would continue to visit his grandmother up to her funeral. Franciszka worked in a branch of the Tńôcza laundries while her husband was the manager of the Krakowski Hotel. Together, they were able to give Banach what was a relatively comfortable life for the time. Contacts between Banach and his father were polite and cordial; though Banach loved his father, he did not show him much warmth or filial affection.

As a child, Banach was introduced to Juliusz Mien, a French intellectual who had moved to Krak√≥w in 1870 and who was a guardian of Maria PŇāowa. Mien guided Banach by teaching him French and supervising his education without charge. Mien likely nurtured Banach's early mathematical skills, and he taught him to speak French so fluently that later in life Banach was able to impress foreign colleagues with his knowledge of the language.

In 1902 Banach, aged 10, enrolled in Krak√≥w's Henryk Sienkiewicz Gymnasium no. IV where he became known as a prodigy. The school specialized in the humanities, including languages such as Latin, Greek, and German as well as subjects such as History and Geography along side some Mathematics. Despite this shortcoming, Banach and his best friend Witold WiŇākosz, a future mathematician, would regularly work on mathematics problems during school breaks and after school. In 1906 Banach, aged 14, was studying higher mathematics and two years later he had started in on several languages, both western and eastern, however he was especially fond of Latin. After obtaining his matura at age 18 in 1910, Banach went with Witold WiŇākosz to Lviv, then the capital of Galicia, intending to enroll in engineering at the Lw√≥w 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.

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 gymnasiums and working in a bookshop. He may have attended lectures at the Jagiellonian University, but little is known of that period in his life.

In 1916, in Krak√≥w's Planty gardens, Banach encountered Professor Hugo Steinhaus, one of the 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, ŇĀucja Braus.

Steinhaus introduced Banach to academic circles and substantially accelerated his career. After Poland regained independence, in 1920 Banach was given an assistantship at Krak√≥w's Jagiellonian University. Steinhaus' backing also allowed him to receive a doctorate without actually graduating from a university. The doctoral thesis, accepted by King John II Casimir University of Lw√≥w and published in 1922, included the basic ideas of functional analysis, 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 Lw√≥w Polytechnic. Initially an assistant to Professor Antoni ŇĀomnicki, in 1927 Banach received his own chair. In 1924 he was also accepted as a member of the Polish Academy of Learning. At the same time, from 1922, Banach also headed the second Chair of Mathematics at University of Lw√≥w.

Young and talented, Banach gathered around him a large group of mathematicians. The group, meeting in the Scottish Caf√©marker, soon gave birth to the "Lw√≥w School of Mathematics." In 1929 the group began publishing its own journal, Studia Mathematica, devoted primarily to Banach's field of study ‚ÄĒ 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, 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.

Following the invasion of Poland by Nazi Germany and the Soviet Unionmarker, Lwów came under the control of the Soviet Unionmarker for almost two years. Banach, from 1939 a corresponding member of the Academy of Sciences of Ukraine, and on good terms with Sovietmarker 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, all universities were closed and Banach, along with many colleagues and his son, was employed as lice feeder at Professor Rudolf Weigl's Typhus 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.

After the Red Army recaptured Lvivmarker in the Lvov‚ÄďSandomierz Offensive of 1944, Banach returned to the University and helped re-establish it after the war years. However, because the Soviets were removing Poles from annexed formerly Polish territories, Banach began preparing to leave the city and settle in Krak√≥wmarker, Polandmarker, where he had been promised a chair at the Jagiellonian University. 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 Lw√≥w. He died on August 31, 1945, aged 53. His funeral at the Lychakiv Cemeterymarker turned into a patriotic demonstration by the Poles who still remained in the city.


  • Rachunek r√≥Ňľniczkowy i caŇākowy, tom I (Differential and Integral Calculus, vol. 1), Lw√≥w, ZakŇāad Narodowy im. OssoliŇĄskich, 1929, 294 pp.
  • Rachunek r√≥Ňľniczkowy i caŇākowy, tom II (Differential and Integral Calculus, vol. 2), Lw√≥w, KsińÖŇľnica-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.
  • Th√©orie des op√©rations lin√©aires, Monografie Matematyczne 1 (Theory of Linear Operations, Mathematical Monographs 1), Warsaw, 1932, vii + 254 pp.
  • Mechanika w zakresie szk√≥Ňā akademickich, Monografie Matematyczne 8 (Mechanics for Academic Schools, Mathematical Monographs 8), Warsaw, Lw√≥w, 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," and proved many fundamental theorems of functional analysis.

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

He was also one of the founders and editors of the journal, Studia Mathematica.


StanisŇāaw Ulam, another mathematician of the Lw√≥w School of Mathematics, in his autobiography, quotes Banach as saying:
"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."

See also


  1. Stefan Banach: Teoria operacji liniowych.
  2. Stefan Banach: Théorie des opérations linéaires (Theory of Linear Operations).
  3. Artyku³y o Lwowie at
  4. Wortal Stefana Banacha at
  5. Tadeusz KrzyŇľewski, as cited in:


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