Stefan Banach | |
---|---|

Born | |

Died | 31 August 1945 53) | (aged

Nationality | Polish |

Alma mater | Technical University of Lwów |

Known for | Banach space Functional analysis Banach algebra Banach measure Banach–Tarski paradox Banach fixed-point theorem Banach–Steinhaus theorem Banach–Mazur theorem Banach–Schauder theorem Hahn–Banach theorem Banach–Alaoglu theorem Surjection of Fréchet spaces |

Awards | Memberships: Academy of Sciences of the Ukrainian SSR, Polish Academy of Learning |

Scientific career | |

Fields | Mathematics |

Institutions | University of Lwów |

Doctoral advisors | Hugo Steinhaus Kazimierz Twardowski |

Doctoral students | Stanisław Mazur |

Other notable students | Józef Schreier Stanislaw Ulam |

**Stefan Banach** (Polish: [ˈstɛfan ˈbanax] ( listen ); 30 March 1892 – 31 August 1945) was a Polish mathematician ^{ [1] } who is generally considered one of the world's most important and influential 20th-century mathematicians. He was the founder of modern functional analysis,^{ [2] } and an original member of the Lwów School of Mathematics. His major work was the 1932 book, *Théorie des opérations linéaires* (Theory of Linear Operations), the first monograph on the general theory of functional analysis.

- Life
- Early life
- Discovery by Steinhaus
- Interbellum
- World War II
- Contributions
- Stefan Banach Medal
- Quotes
- See also
- Notes
- Further reading
- References
- External links

Born in Kraków, Banach attended IV Gymnasium, a secondary school, and worked on mathematics problems with his friend Witold Wilkosz. After graduating in 1910, Banach moved to Lwów (today called Lviv) . However, during World War I Banach returned to Kraków, where he befriended Hugo Steinhaus. After Banach solved some mathematics problems that Steinhaus considered difficult, they published their first joint work. In 1919, with several other mathematicians, Banach formed a mathematical society. In 1920 he received an assistantship at the Lwów Polytechnic. He soon became a professor at the Polytechnic, and a member of the Polish Academy of Learning. He organized the "Lwów School of Mathematics". Around 1929 he began writing his *Théorie des opérations linéaires*.

After the outbreak of World War II, in September 1939, Lwów was taken over by the Soviet Union. Banach became a member of the Academy of Sciences of Ukraine and was dean of Lwów University's Department of Mathematics and Physics. In 1941, when the Germans took over Lwów, all institutions of higher education were closed to Poles. 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. While the job carried the risk of infection with typhus, it protected him from being sent to slave labor in Germany and from other forms of repression. When the Soviets recaptured Lwów in 1944, Banach reestablished the University. However, because the Soviets were removing Poles from Soviet-annexed formerly-Polish territories, Banach prepared to return to Kraków. Before he could do so, he died in August 1945, having been diagnosed seven months earlier with lung cancer.

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 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^{ [3] } and was subsequently baptised by his father, while his mother abandoned him upon this event and her identity is ambiguous.^{ [4] }^{ [5] } Banach's parents were Stefan Greczek and Katarzyna Banach, both natives of the Podhale region.^{ [6] }^{ [7] } Greczek was a soldier in the Austro-Hungarian Army stationed in Kraków. Little is known about Banach's mother.^{ [8] } According to his baptismal certificate, she was born in Borówna and worked as a domestic help.^{ [7] }

Unusually, Stefan's surname was his mother's instead of his father's, though he received his father's given name, Stefan. Since Stefan Greczek 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.^{ [9] } Stefan spent the first few years of his life with his grandmother, but when she took ill Greczek arranged for his son to be raised by Franciszka Płowa and her niece Maria Puchalska in Kraków. Young Stefan would regard Franciszka as his foster mother and Maria as his older sister.^{ [10] } In his early years Banach was tutored by Juliusz Mien, a French intellectual and friend of the Płowa family, who had emigrated to Poland and supported himself with photography and translations of Polish literature into French. Mien taught Banach French and most likely encouraged him in his early mathematical pursuits.^{ [11] }

In 1902 Banach, aged 10, enrolled in Kraków's *IV Gymnasium* (also known as the * Goetz Gymnasium *). While the school specialized in the humanities, Banach and his best friend Witold Wiłkosz (also a future mathematician) spent most of their time working on mathematics problems during breaks and after school.^{ [12] } Later in life Banach would credit Dr. Kamil Kraft, the mathematics and physics teacher at the gymnasium with kindling his interests in mathematics.^{ [13] } While generally Banach was a diligent student he did on occasion receive low grades (he failed Greek during his first semester at the gymnasium) and would later speak critically of the school's math teachers.^{ [14] }

After obtaining his * matura * (high school degree) at age 18 in 1910, Banach moved to Lwów (today called Lviv) with the intention of studying at the Lwów Polytechnic. He initially chose engineering as his field of study since at the time he was convinced that there was nothing new to discover in mathematics.^{ [15] } At some point he also attended Jagiellonian University in Kraków on a part-time basis. 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.^{ [16] }

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, where he spent the rest of the war. He made his living as a tutor at the local gymnasiums, worked in a bookstore and as a foreman of road building crew. 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.^{ [17] }

In 1916, in Kraków's * Planty * gardens, Banach encountered Professor Hugo Steinhaus, one of the renowned mathematicians of the time. According to Steinhaus, while he was strolling through the gardens he was surprised to overhear the term *"Lebesgue integral"* (Lebesgue integration was at the time still a fairly new idea in mathematics) and walked over to investigate. As a result, he met Banach, as well as Otto Nikodym.^{ [18] } Steinhaus became fascinated with the self-taught young mathematician. The encounter resulted in a long-lasting collaboration and friendship. In fact, soon after the encounter Steinhaus invited Banach to solve some problems he had been working on but which had proven difficult. Banach solved them within a week and the two soon published their first joint work (*On the Mean Convergence of Fourier Series*). 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.^{ [19] } The society was officially founded on 2 April 1919. It was also through Steinhaus that Banach met his future wife, Łucja Braus.

Steinhaus introduced Banach to academic circles and substantially accelerated his career. After Poland regained independence in 1918, Banach was given an assistantship at the Lwów Polytechnic. 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 Lwów in 1920 ^{ [20] } and published in 1922,^{ [21] } 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 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. At the same time, from 1922, Banach also headed the second Chair of Mathematics at University of Lwów.

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,^{ [22] } the following year it was also translated into French and gained wider recognition in European academic circles.^{ [23] } The book was also the first in a long series of mathematics monographs edited by Banach and his circle. In 17 June 1924 Banach become a correspondence member of the Polish Academy of Sciences and Fine Arts in Kraków.

Following the invasion of Poland by Nazi Germany and the Soviet Union, Lwów came under the control of the Soviet Union for almost two years. Banach, from 1939 a corresponding member of the Academy of Sciences of Ukraine, and on good terms with Soviet mathematicians,^{ [8] } had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities.^{ [24] } Following the German takeover of Lwów in 1941 during 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 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 from annexed formerly Polish territories, Banach began preparing to leave the city and settle in Kraków, Poland, where he had been promised a chair at the Jagiellonian University.^{ [8] } He was also considered a candidate for Minister of Education of Poland.^{ [25] } In January 1945, however, he was diagnosed with lung cancer and was allowed to stay in Lwów. He died on 31 August 1945, aged 53. His funeral at the Lychakiv Cemetery was attended by hundreds of people.^{ [25] }

Banach's dissertation, completed in 1920 and published in 1922, formally axiomatized the concept of a complete normed vector space and laid the foundations for the area of functional analysis. In this work Banach called such spaces *"class E-spaces"*, but in his 1932 book, *Théorie des opérations linéaires*, he changed terminology and referred to them as *"spaces of type B"*, which most likely contributed to the subsequent eponymous naming of these spaces after him.^{ [26] } The theory of what came to be known as Banach spaces had antecedents in the work of the Hungarian mathematician Frigyes Riesz (published in 1916) and contemporaneous contributions from Hans Hahn and Norbert Wiener.^{ [20] } For a brief period in fact, complete normed linear spaces were referred to as "Banach–Wiener" spaces in mathematical literature, based on terminology introduced by Wiener himself. However, because Wiener's work on the topic was limited, the established name became just *Banach spaces*.^{ [26] }

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 did not require linearity of the space, and applied to any complete Cauchy space (in particular to any complete metric space).^{ [20] }

The Hahn–Banach theorem is one of the fundamental theorems of functional analysis.^{ [20] }

In 1992 the Institute of Mathematics of the Polish Academy of Sciences established a special Stefan Banach medal for outstanding achievements in mathematical sciences.^{ [27] }

Stanislaw Ulam, another mathematician of the Lwów School of Mathematics, in his autobiography, quotes Banach as saying:

- "Good mathematicians see analogies. Great mathematicians see analogies between analogies."
^{ [28] }

Hugo Steinhaus said of Banach:

- "Banach was my greatest scientific discovery."
^{ [29] }

- Closed range theorem
- International Stefan Banach Prize
- List of Poles
- List of Polish mathematicians – Wikipedia list article
- List of things named after Stefan Banach – Wikipedia list article

- ↑ https://www.britannica.com/biography/Stefan-Banach, “Stefan Banach POLISH MATHEMATICIAN”
- ↑ https://www.britannica.com/biography/Stefan-Banach, “Polish mathematician who founded modern functional analysis and helped develop the theory of topological vector spaces.”
- ↑ "Home Page of Stefan Banach".
*kielich.amu.edu.pl*. Retrieved 19 August 2017. - ↑ "Banach biography".
*www-history.mcs.st-andrews.ac.uk*. Retrieved 19 August 2017. - ↑ Peter Stachura,
*Poland in the Twentieth Century*, Springer (1999), p. 51 - ↑ Waksmundzka-Hajnos 2006, p. 16
- 1 2 Duda, Roman (2009). "Facts and Myths about Stefan Banach" (PDF).
*Newsletter of the European Mathematical Society*. EMS (71): 29. - 1 2 3 O'Connor & Robertson 2000
- ↑ Kałuża 1996 , pp. 2–4
- ↑ Kałuża 1996 , pp. 1–3
- ↑ Kałuża 1996 , p. 3
- ↑ Kałuża 1996 , p. 137
- ↑ Jakimowicz & Miranowicz 2007, p. 4
- ↑ Kałuża 1996 , pp. 3–4
- ↑ Jakimowicz & Miranowicz 2007, p.5
- ↑ Kałuża 1996 , p. 13
- ↑ Kałuża 1996 , p. 16
- ↑ Jakimowicz & Miranowicz 2007, p. 6
- ↑ Kałuża 1996, p. 23
- 1 2 3 4 Jahnke 2003, p. 402
- ↑ Stefan Banach (1922). "Sur les opérations dans les ensembles abstraits et leur application aux équations integrals (On operations in the abstract sets and their application to integral equations)".
*Fundamenta Mathematicae*(in French and Polish).**3**. - ↑ Stefan Banach:
*Teoria operacji liniowych*. - ↑ Stefan Banach:
*Théorie des opérations linéaires*(in French; Theory of Linear Operations). - ↑ Urbanek 2002
- 1 2 James 2003, p. 384
- 1 2 MacCluer 2008 , p. 6
- ↑ Institute of Mathematics: Stefan Banach Medal Polish Academy of Sciences
- ↑ National Research Council of the National Academies (25 March 2014).
*Developing a 21st Century Global Library for Mathematics Research*. National Academies Press. p. 35. ISBN 9780309298513 . Retrieved 28 March 2018. - ↑ Strick, Heinz Klaus (2011). "Stefan Banach (March 30, 1892 – August 8, 1945)".
*Mathematics in Europe*. Translated by Kramer, David. European Mathematical Society. Retrieved 28 March 2018.

- Banach, Stefan (1932).
*Théorie des Opérations Linéaires*[*Theory of Linear Operations*](PDF). Monografie Matematyczne (in French).**1**. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 11 January 2014. Retrieved 11 July 2020.

**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 functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

**Kazimierz Kuratowski** was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.

**Władysław Hugo Dionizy Steinhaus** was a Jewish-Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz University in Lwów, where he helped establish what later became known as the Lwów School of Mathematics. He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach–Steinhaus theorem. After World War II Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war.

The **Scottish Café** was a café in Lwów, Poland where, in the 1930s and 1940s, mathematicians from the Lwów School of Mathematics collaboratively discussed research problems, particularly in functional analysis and topology.

**Stanisław Mieczysław Mazur** was a Polish mathematician and a member of the Polish Academy of Sciences.

The **Lwów school of mathematics** was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland. The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the journal *Studia Mathematica*, founded in 1929. The school was renowned for its productivity and its extensive contributions to subjects such as point-set topology, set theory and functional analysis. The biographies and contributions of these mathematicians were documented in 1980 by their contemporary Kazimierz Kuratowski in his book *A Half Century of Polish Mathematics: Remembrances and Reflections*.

**Antoni Marian Łomnicki** was a Polish mathematician.

Education has been of prime interest to Poland's rulers since the early 12th century. The catalog of the library of the Cathedral Chapter in Kraków dating from 1110 shows that Polish scholars already then had access to western European literature. In 1364, King Kazimierz the Great founded the Cracow Academy, which would become one of the great universities of Europe. The Polish people have made considerable contributions in the fields of science, technology and mathematics. The list of famous scientists in Poland begins in earnest with the polymath, astronomer and mathematician Nicolaus Copernicus, who formulated the heliocentric theory and made an important contribution to the scientific revolution.

The **Scottish Book** was a thick notebook used by mathematicians of the Lwów School of Mathematics in Poland for jotting down problems meant to be solved. The notebook was named after the "Scottish Café" where it was kept.

**Feliks Barański** (1915-2006) was a Polish mathematician and an active member of the so-called Lwów School of Mathematics. Born May 1915 in Lwów, Austria-Hungary, he joined the circle of young, talented mathematicians formed around Stefan Banach and Hugo Steinhaus. During the period of German occupation of his hometown he made his living as a lice feeder in the institute of Rudolf Weigl. Expelled from Lwow after the war, he settled in Kraków, where he joined the local Kraków University of Technology. He was also admitted into the Polish Mathematics Society.

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.

In functional analysis, the **open mapping theorem**, also known as the **Banach–Schauder theorem**, is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.

**Joram Lindenstrauss** was an Israeli mathematician working in functional analysis. He was a professor of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel.

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.

**Mikhail Iosiphovich Kadets** was a Soviet-born Jewish mathematician working in analysis and the theory of Banach spaces.

A **louse-feeder** was a job in interwar and Nazi-occupied Poland, at the *Lviv Institute for Study of Typhus and Virology* and the associated Institute in Kraków, Poland. Louse-feeders were human sources of blood for lice infected with typhus, which were then used to research possible vaccines against the disease.

**Józef Schreier** was a Polish mathematician of Jewish origin, known for his work in functional analysis, group theory and combinatorics. He was a member of the Lwów School of Mathematics and a victim of the Holocaust.

**Meier "Maks" Eidelheit** (6 July 1910, Janów – March 1943) was a Polish mathematician belonging to the Lwów School of Mathematics who worked in Lwów and perished in the Holocaust.

- Jahnke, Hans Niels (2003).
*A History of Analysis*. American Mathematical Society. ISBN 0821826239. - Jakimowicz, E.; Miranowicz, A., eds. (2007).
*Stefan Banach - Remarkable life, Brilliant mathematics*. Gdańsk University Press and Adam Mickiewicz University Press. ISBN 978-83-7326-451-9. - James, Ioan (2003).
*Remarkable Mathematicians: From Euler to von Neumann*. Cambridge University Press. ISBN 0521520940. - Kałuża, Roman (1996).
*Through a Reporter's Eyes: The Life of Stefan Banach*. Translated by Wojbor Andrzej Woyczyński and Ann Kostant. Birkhäuser. ISBN 0-8176-3772-9. - Kosiedowski, Stanisław. "Stefan Banach".
*Mój Lwów*. Retrieved 20 May 2008. - O'Connor, John J.; Robertson, Edmund F. (2000). "Stefan Banach".
*MacTutor History of Mathematics archive*. University of St. Andrews. Retrieved 19 August 2012. - Siegmund-Schultze, Reinhard (2003). Jahnke, Hans Niels (ed.).
*A History of Analysis*. American Mathematical Society. ISBN 0-8218-2623-9. - MacCluer, Barbara (2008).
*Elementary Functional Analysis*. Springer. ISBN 978-0387855288. - Urbanek, Mariusz (April 2002). "Geniusz: gen i już".
*Polityka*.**8**(2348). - Waksmundzka-Hajnos, Monika (2006). "Wspomnienie o Stefanie Greczku".
*Focus*. Gdańsk University (11).

- Page devoted to Stefan Banach
- Stefan Banach at the Mathematics Genealogy Project
- Works by or about Stefan Banach in libraries ( WorldCat catalog)

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