Editing Stefan Banach

{{Short description|Polish mathematician (1892–1945)}}
{{Use dmy dates|date=March 2018}}
{{Infobox scientist
|name = Stefan Banach
|image = عالم الرياضيات البولندى ستيفان بناخ.jpg
|image_size = 200px
|caption = 
|birth_date = {{Birth date|df=yes|1892|3|30}}
|birth_place = [[Kraków]], [[Austria-Hungary]]<br>(today Poland)
|death_date = {{Death date and age|df=yes|1945|8|31|1892|3|30}}
|death_place = [[Lviv]], [[Ukrainian SSR]], [[Soviet Union]] (today Ukraine)
|field = [[Mathematics]]
|work_institutions = [[Lviv University|University of Lwów]]
|alma_mater = [[Lviv Polytechnic|Technical University of Lwów]]
|doctoral_advisors = [[Hugo Steinhaus]]<br>[[Kazimierz Twardowski]]
|doctoral_students = [[Stanisław Mazur]]
|notable_students = [[Józef Schreier]]<br>[[Stanislaw Ulam]]
|known_for = [[Banach space]]<br />[[Functional analysis]]<br />[[Banach algebra]]<br />[[Banach measure]]<br />[[Banach–Tarski paradox]]<br />[[Banach fixed-point theorem]]<br />[[Banach–Steinhaus theorem]]<br />[[Banach–Mazur theorem]]<br />[[Banach–Schauder theorem]]<br />[[Hahn–Banach theorem]]<br />[[Banach–Alaoglu theorem]]<br />[[Banach–Stone theorem]]<br />[[Banach manifold]]<br />[[Banach bundle]]<br />[[Surjection of Fréchet spaces]]
|author_abbrev_bot = 
|author_abbrev_zoo = 
|influences = 
|influenced = 
|prizes = Membership: [[Polish Academy of Learning]]
|religion = 
|footnotes = 
|signature =
}}

'''Stefan Banach''' ({{small|Polish:}} {{IPAc-pl|AUD|Pl-Stefan_Banach.ogg|'|s|t|e|f|a|n|-|'|b|a|n|a|x}}; 30 March 1892 – 31 August 1945) was a Polish [[mathematician]]<ref name=EB /> who is generally considered one of the 20th century's most important and influential mathematicians.{{sfn|Pitici|2019|p=23}} He was the founder of modern [[functional analysis]],<ref name=EB /> 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.{{sfn|Chemla| Chorla| Rabouin|2016|pp=224, 225, 237}}

Born in [[Kraków]] to a family of [[Gorals|Goral]] descent, Banach showed a keen interest in mathematics and engaged in solving [[mathematical problem]]s during school [[Recess (break)|recess]]. After completing his secondary education, he befriended [[Hugo Steinhaus]], with whom he established the [[Polish Mathematical Society]] in 1919 and later published the scientific journal ''[[Studia Mathematica]]''. In 1920, he received an assistantship at the [[Lwów Polytechnic]], subsequently becoming a professor in 1922 and a member of the [[Polish Academy of Learning]] in 1924. Banach was also a co-founder of the [[Lwów School of Mathematics]], a [[school of thought]] comprising some of the most renowned Polish mathematicians of the [[interwar period]] (1918–1939).

Some of the notable [[mathematical]] [[concept]]s that bear Banach's name include [[Banach space]]s, [[Banach algebra]]s, [[Banach measure]]s, 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]].

==Life==

===Early life===
Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in [[Kraków]], then part of the [[Austria-Hungary|Austro-Hungarian Empire]], into a [[Gorals|Góral]] [[Roman Catholic]] family,<ref name=SBHomePage /> and was subsequently baptised by his father.<ref name=MacTutor />{{sfn|Stachura|1999|p=51}} Banach's parents were Stefan Greczek and Katarzyna [[Banach]], both natives of the [[Podhale]] region.{{sfn|Waksmundzka-Hajnos|2006|p=16}}{{sfn|Duda|2009|p=29}} Greczek was a soldier in the [[Austro-Hungarian Army]] stationed in Kraków. Little is known about Banach's mother.<ref name=MacTutor /> According to his [[Baptism|baptismal certificate]], she was born in [[Borówna]] and worked as a domestic helper.{{sfn|Duda|2009|p=29}}

Unusually, Stefan's surname was his mother's instead of his father's, though he received his father's given name, Stefan. Military regulations did not permit soldiers of Stefan Greczek's rank to [[marriage|marry]]; he was a [[private (rank)|private]] and as the mother was too poor to support the child, the couple decided that he should be reared by family and friends.<ref name="Kałuża-2–4">{{Harvnb|Kałuża|1996|pp=2–4}}</ref> Stefan spent the first few years of his life with his grandmother, but when she was taken ill, Greczek arranged for his son to be raised by Franciszka Płowa and her niece Maria Puchalska in Kraków. Young Stefan came to regard Franciszka as his foster mother and Maria as his older sister.<ref name="Kałuża-1-3">{{Harvnb|Kałuża|1996|pp=1–3}}</ref> 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.<ref name="Kałuża-3">{{Harvnb|Kałuża|1996|p=3}}</ref>

In 1902, Banach, aged 10, enrolled in Kraków's ''IV Gymnasium'' (also known as the ''[[Jan Albin Goetz|Goetz Gymnasium]]''). While the school specialized in the [[humanities]], Banach and his best friend [[Witold Wilkosz|Witold Wiłkosz]] (also a future mathematician) spent most of their time working on mathematics problems during breaks and after school.<ref name="Kałuża-137">{{Harvnb|Kałuża|1996|p=137}}</ref> Later in life Banach credited Dr. Kamil Kraft, the mathematics and physics teacher at the school, with kindling his interests in mathematics.<ref name="rem4"/> While Banach was a diligent student he did, on occasion, receive low grades (he failed Greek during his first semester at the school) and later spoke critically of the school's math teachers.<ref name="Kałuża-3-4">{{Harvnb|Kałuża|1996|pp=3–4}}</ref>

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.<ref name="rem5"/> 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.<ref name="Kałuża-13">{{Harvnb|Kałuża|1996|p=13}}</ref>

When [[World War I]] broke out, Banach was excused from military service due to his left-handedness and poor vision. When the [[Imperial Russian Army|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 schools, worked in a bookstore and as a foreman of a road building crew. He attended some lectures at the [[Jagiellonian University]] at that time, including those of the famous Polish mathematicians [[Stanisław Zaremba (mathematician)|Stanisław Zaremba]] and [[Kazimierz Żorawski]], but little is known of that period of his life.<ref name="Kałuża-16">{{Harvnb|Kałuża|1996|p=16}}</ref>

===Discovery by Steinhaus===
[[File:Otto Nikodym Stefan Banach Memorial Bench Krakow Poland.jpg|thumb|200px|[[Otto Nikodym]] and Stefan Banach Memorial Bench in [[Kraków]], [[Poland]] (sculpted by Stefan Dousa)]]
In 1916, in Kraków's ''[[Planty Park|Planty]]'' park, 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]].<ref name="rem6" /> 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{{sfn | Ciesielska | Maligranda | 2019 | pp=57–108}} and [[Włodzimierz Stożek]]) also established a mathematical society, which eventually became the [[Polish Mathematical Society]].<ref name="Kaluza-23"/> The society was officially founded on 2 April 1919. It was also through Steinhaus that Banach met his future wife, Łucja Braus.

===Interbellum===
[[File:Lwow - Kawiarnia Szkocka.jpg|thumb|200px|[[Scottish Café]], meeting place of many famous Lwów mathematicians]]
Steinhaus introduced Banach to academic circles and substantially accelerated his career. After [[Second Polish Republic|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 [[Lviv University|King John II Casimir University of Lwów]] in 1920<ref name="Jahnke"/> and published in 1922,<ref name="Fundamenta"/> included the basic ideas of functional analysis, which was soon to become an entirely new branch of mathematics. In his dissertation, written in 1920, he axiomatically defined what is today called a Banach space.<ref>{{cite web | url=https://krakow1.one/uk/eternal/matematychnyj-mist-mizh-krakovom-i-lvovom-yak-stefan-banah-stav-odnym-iz-najvelychnishyh-matematykiv-stolittya-2181 | title=Математичний міст між Краковом і Львовом: як Стефан Банах став одним із найвеличніших математиків століття - krakow1.one | date=16 November 2022 }}</ref> 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 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 [[Lviv University|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 space|linear]]-[[metric space]]. First published in Polish in 1931,<ref name="Teoria"/> the next year it was also translated into French and gained wider recognition in European academic circles.<ref name="Theorie"/> The book was also the first in a long series of mathematics monographs edited by Banach and his circle. On 17 June 1924, Banach became a [[Corresponding member|correspondence member]] of the [[Polish Academy of Learning|Polish Academy of Sciences and Fine Arts]] in Kraków.

===World War II===
[[File:Cmentarz-Lyczakowski-Grob Banacha.jpg|thumb|left|200px|Banach's grave, [[Lychakiv Cemetery]], [[Lviv]] (''[[Lwów]]'', in [[Polish language|Polish]])]]
After 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 Union|Soviet]] mathematicians,<ref name=MacTutor /> had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities.<ref name="Urbaniak"/> In 1940 he was appointed by the Soviets as a member of the [[Lviv City Council]]. After 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 a [[Feeder of lice|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 Soviet [[Red Army]] recaptured [[Lwów|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 [[Polish population transfers (1944–1946)|deporting Poles]] from annexed formerly Polish [[Kresy|eastern territories]], Banach began preparing to leave the city and settle in [[Kraków]], Poland, where he had been promised a chair at the [[Jagiellonian University]].<ref name=MacTutor /> He was also considered a candidate for Minister of Education of Poland.<ref name="remark"/> In January 1945, he was diagnosed with [[lung cancer]] and was permitted to stay in Lwów. He died on 31 August 1945, aged 53. His funeral at the [[Lychakiv Cemetery]] (''Cmentarz Łyczakowski'') was attended by hundreds of people.<ref name="remark"/>

==Contributions==
Banach's dissertation, completed in 1920 and published in 1922, formally axiomatized the concept of a [[complete metric space|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 [[eponym]]ous naming of these spaces after him.<ref name="Mac">{{Harvnb|MacCluer|2008|p=6}}</ref> The theory of what came to be known as [[Banach space]]s had antecedents in the work of the Hungarian mathematician [[Frigyes Riesz]] (published in 1916) and contemporaneous contributions from [[Hans Hahn (mathematician)|Hans Hahn]] and [[Norbert Wiener]].<ref name="Jahnke"/> 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''.<ref name="Mac"/>

Likewise, [[Banach fixed-point theorem|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 [[L. E. J. Brouwer|Brouwer]] and [[Poincaré–Birkhoff theorem|Poincaré and Birkhoff]]). The theorem did not require linearity of the [[Space (mathematics)|space]], and applied to any complete [[Cauchy space]] (in particular to any complete metric space).<ref name="Jahnke"/>

[[File:Banach-Tarski Paradox.svg|thumb|upright=1.5|Decomposition of a ball into two identical balls – the [[Banach–Tarski paradox]]]]
The [[Hahn–Banach theorem]] is one of the fundamental theorems of functional analysis.<ref name="Jahnke"/> Further theorems related to Banach are:

* [[Banach–Tarski paradox]]
* [[Uniform boundedness principle|Banach–Steinhaus theorem]]
* [[Banach–Alaoglu theorem]]
* [[Banach–Stone theorem]]

==Recognition==
In 1946, the [[Stefan Banach Prize]] (Polish: ''Nagroda im. Stefana Banacha'') was established by the [[Polish Mathematical Society]]. In 1992, the [[Institute of Mathematics of the Polish Academy of Sciences]] established a special [[Stefan Banach Medal]] for outstanding achievements in mathematical sciences.<ref>[https://www.impan.pl/en/events/awards/stefan-banach-medal Institute of Mathematics: Stefan Banach Medal] Polish Academy of Sciences</ref> Since 2009, the [[International Stefan Banach Prize]] has been conferred by the Polish Mathematical Society to mathematicians for best doctoral dissertations in the mathematical sciences with the objective to "promote and financially support the most promising young researchers".<ref>{{cite web|url=https://banachprize.org/2009/04/pierwszy-laureat-the-international-banach-prize/?lang=pl |archive-url=https://web.archive.org/web/20150617142955/https://banachprize.org/2009/04/pierwszy-laureat-the-international-banach-prize/?lang=pl |archive-date=2015-06-17  |language=pl |title=PIERWSZY LAUREAT "THE INTERNATIONAL BANACH PRIZE" |access-date=11 August 2022}}</ref>

Stefan Banach is the patron of a number of schools and streets including in [[Warsaw]], [[Lviv]], [[Świdnica]], [[Toruń]] and [[Jarosław]].

In 2001, a minor planet [[16856 Banach]], discovered by [[Paul Comba]] in 1997, was named after him.<ref>{{cite book |last=Urbanek |first=Mariusz |author-link= |date=2014 |title=Genialni. Lwowska Szkoła Matematyczna |url= |location=Warsaw |publisher=Wydawnictwo Iskry |page= |isbn=978-83-244-0381-3}}</ref>

In 2012, the [[National Bank of Poland]] celebrated the mathematician's achievements by issuing a series of commemorative coins designed by Robert Kotowicz (golden 200-[[Polish zloty|zloty]] coin, silver 10-zloty coin and [[Nordic Gold]] 2-zloty coin).<ref>{{cite web| url = https://www.nbp.pl/banknoty_i_monety/monety_okolicznosciowe/2012/2012_05___stefan_banach_pl.pdf |language=pl |title=Narodowy Bank Polski. Monety. |access-date=11 August 2022}}</ref>

In 2016, a commemorative bench featuring Banach and [[Otto Nikodym]] was unveiled in Kraków's [[Planty Park (Kraków)|Planty Park]] on the 100th anniversary of the conversation the two mathematicians held when they first met [[Hugo Steinhaus]], which proved instrumental in the development of his scientific career.<ref>{{cite web|url=http://www.radiokrakow.pl/wiadomosci/krakow/setna-rocznica-najslynniejszej-matematycznej-dyskusji-na-krakowskich-plantach/wiadomosci/krakow/setna-rocznica-najslynniejszej-matematycznej-dyskusji-na-krakowskich-plantach/ |archive-url=https://web.archive.org/web/20161019011436/http://www.radiokrakow.pl/wiadomosci/krakow/setna-rocznica-najslynniejszej-matematycznej-dyskusji-na-krakowskich-plantach/wiadomosci/krakow/setna-rocznica-najslynniejszej-matematycznej-dyskusji-na-krakowskich-plantach/ |archive-date=2016-10-19  |language=pl |title=Setna rocznica najsłynniejszej matematycznej dyskusji na krakowskich Plantach |access-date=11 August 2022}}</ref>

In 2021, one of the episodes of Polish documentary TV series ''Geniusze i marzyciele'' (Geniuses and Dreamers) aired on [[TVP1]] and TVP Dokument channels was devoted to Stefan Banach.<ref>{{cite web| url = https://centruminformacji.tvp.pl/51985536/geniusze-i-marzyciele-nowy-serial-dokumentalny-w-tvp1 |language=pl |title="Geniusze i marzyciele" – nowy serial dokumentalny w TVP1 |access-date=11 August 2022}}</ref>

In 2022, [[Google Doodle]] commemorated the 100th anniversary of Banach receiving his title of professor.<ref>{{cite web| url = https://www.scotsman.com/news/people/stefan-banach-google-doodle-celebrates-polish-mathematician-3778341 |title=Stefan Banach: Google Doodle celebrates Polish mathematician |date=22 July 2022 |access-date=11 August 2022}}</ref>

==Quotes==
[[File:Stefan banach monumento krakow 2007.jpeg|thumb|150px|Banach monument, [[Kraków]]]]
[[Stanislaw Ulam]], another mathematician of the [[Lwów School of Mathematics]], in his autobiography, quotes Banach as saying:
{{blockquote|"Good mathematicians see analogies between theorems or theories, the very best ones see analogies between analogies."<ref>{{cite book|isbn=9780684143910|title=Adventures of a Mathematician |last1=Ulam |first1=Stanislaw M. |year=1976 |page=203 |publisher=Scribner }}</ref>}}

[[Hugo Steinhaus]] said of Banach:
{{blockquote|"Banach was my greatest scientific discovery."<ref>{{Cite news |date=2016-11-03 |title=Stefan Banach (March 30, 1892 – August 8, 1945) – Mathematics in Europe |url=http://mathematics-in-europe.eu/?p=250 |access-date=2024-01-19 |last=Strick |first=Heinz Klaus |translator-first=David |translator-last=Kramer |website=Mathematics in Europe |publisher=European Mathematical Society |archive-url=https://web.archive.org/web/20161103020818/http://mathematics-in-europe.eu/?p=250 |archive-date=3 November 2016 |language=en-us}}</ref>}}

== See also ==
* [[Closed range theorem]]
* [[Polish Mathematical Society#International Stefan Banach Prize|International Stefan Banach Prize]]
* [[List of Poles#Mathematics|List of Poles]]
* [[List of Polish mathematicians]]
* [[List of things named after Stefan Banach]]
* [[Timeline of Polish science and technology]]

== References ==
===Citations===
{{Reflist|30em|refs=

<ref name=MacTutor>{{MacTutor|id=Banach|title=Stefan Banach}}</ref>
<ref name=SBHomePage>{{Cite web|url=http://kielich.amu.edu.pl/Stefan_Banach/e-biography.html|title=Home Page of Stefan Banach|website=kielich.amu.edu.pl|access-date=19 August 2017}}</ref>

<ref name=EB>{{cite web|url=https://www.britannica.com/biography/Stefan-Banach|title=Stefan Banach - Polish Mathematician|website=britannica.com|date=27 August 2023 }}</ref>

<ref name="Fundamenta">{{cite journal|author =Stefan Banach|year =1922 |title =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)|journal =Fundamenta Mathematicae|volume =3|language=fr, pl}}</ref>

<ref name="Jahnke">{{Harvnb|Jahnke|2003|p=402}}</ref>

<ref name="remark">{{Harvnb|James|2003|p=384}}</ref>

<ref name="Kaluza-23">{{harvnb|Kałuża|1996|p=23}}</ref>

<ref name="rem4">{{harvnb|Jakimowicz|Miranowicz|2007}}, p. 4</ref>

<ref name="rem5">{{harvnb|Jakimowicz|Miranowicz|2007}}, p.5</ref>

<ref name="rem6">{{harvnb|Jakimowicz|Miranowicz|2007}}, p. 6</ref>

<ref name="Teoria">Stefan Banach: ''Teoria operacji liniowych''.</ref>

<ref name="Theorie">Stefan Banach: ''Théorie des opérations linéaires'' (in French; Theory of Linear Operations).</ref>

<ref name="Urbaniak">{{Harvnb|Urbanek|2002}}</ref>

}}

=== Sources ===
{{refbegin|2|indent=yes}}
*{{cite book |first1=Karine|last1= Chemla|first2= Renaud|last2= Chorla|first3= David|last3= Rabouin |date=2016 |title=The Oxford Handbook of Generality in Mathematics and the Sciences |url= |location=Oxford |publisher=University Press  |isbn=9780198777267}}
* {{cite journal | last1=Ciesielska | first1=Danuta | last2=Maligranda | first2=Lech | title=Alfred Rosenblatt (1880–1947). Polish–Peruvian mathematician | journal=Banach Center Publications | publisher=Institute of Mathematics, Polish Academy of Sciences | volume=119 | year=2019 | issn=0137-6934 | doi=10.4064/bc119-4 | pages=57–108| s2cid=213021027 | doi-access=free }}
*{{cite journal|last1=Duda|first1=Roman|title=Facts and Myths about Stefan Banach|journal=Newsletter of the European Mathematical Society|date=2009|issue=71|page=29|url=https://www.ems-ph.org/journals/newsletter/pdf/2009-03-71.pdf|publisher=EMS}}
* {{cite book | title=A History of Analysis | publisher=American Mathematical Society | last=Jahnke |first= Hans Niels | year=2003 | isbn=0821826239}}
* {{Cite book|editor1-first=E.|editor1-last=Jakimowicz|editor2-first=A.|editor2-last=Miranowicz |title=Stefan Banach - Remarkable life, Brilliant mathematics |publisher=Gdańsk University Press and Adam Mickiewicz University Press |year=2007 |isbn=978-83-7326-451-9 |url =http://kielich.amu.edu.pl/Stefan_Banach/e-jakimowicz_miranowicz.html }}
* {{cite book | title=Remarkable Mathematicians: From Euler to von Neumann | url=https://archive.org/details/remarkablemathem0000jame | url-access=registration | publisher=Cambridge University Press | author-link=Ioan James |last=James |first=Ioan | year=2003 | isbn=0521520940}}
* {{Cite book |first=Roman |last=Kałuża |title=Through a Reporter's Eyes: The Life of Stefan Banach |publisher=Birkhäuser |year=1996 |isbn=0-8176-3772-9|url =https://books.google.com/books?id=i3ZrfMinChkC&q=Stefan+Banach |others =Translated by Wojbor Andrzej Woyczyński and Ann Kostant }}
* {{cite web |last=Kosiedowski|first=Stanisław |title=Stefan Banach|work=Mój Lwów|url=http://www.lwow.com.pl/m.htm|access-date=20 May 2008}}
* {{cite book|editor-first=Hans Niels|editor-last=Jahnke|first=Reinhard|last=Siegmund-Schultze|title=A History of Analysis|publisher=American Mathematical Society|year=2003|isbn=0-8218-2623-9|url=https://books.google.com/books?id=CVRZEXFVsZkC&q=A+History+of+Analysis}}
* {{cite book | title=Elementary Functional Analysis | publisher=Springer | last=MacCluer |first= Barbara | year=2008 | isbn=978-0387855288}}
* {{cite journal | url=http://www.lwow.com.pl/g8.htm | title=Geniusz: gen i już | last=Urbanek |first=Mariusz | journal=Polityka |date=April 2002 | volume=8 | issue=2348}}
*{{cite book |last=Pitici |first=Mircea |date=2019 |title=The Best Writing on Mathematics 2019 |url=https://books.google.com/books?id=RcmXDwAAQBAJ&pg=PA23 |location=Princeton |publisher=University Press  |isbn=978-0691198354}}
*{{cite book|author-link=Peter Stachura|first=Peter |last=Stachura|title=Poland in the Twentieth Century|publisher= Springer|date=1999}}
* {{cite journal | url=http://kielich.amu.edu.pl/Stefan_Banach/greczek.html | title=Wspomnienie o Stefanie Greczku | last=Waksmundzka-Hajnos |first= Monika | publisher = Gdańsk University| journal=Focus | year=2006 | issue=11 }}
{{refend}}

=== Further reading ===
{{refbegin}}
* {{Banach Théorie des Opérations Linéaires}} <!-- {{sfn | Banach | 1932 | p=}} -->
{{refend}}

==External links==
{{Sister project links| wikt=no | commons=no | b=no | n=no | q=Stefan Banach | s=no | v=no | voy=no | species=no | d=no}}

* [http://kielich.amu.edu.pl/Stefan_Banach/e-index.html Page devoted to Stefan Banach]
* {{MathGenealogy|id=12681}}

{{Authority control}}

{{DEFAULTSORT:Banach, Stefan}}
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[[Category:Polish Gorals]]
[[Category:Polish male non-fiction writers]]
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[[Category:Burials at Lychakiv Cemetery]]
[[Category:Deaths from lung cancer in Poland]]
[[Category:Functional analysts]]
[[Category:Ukrainian Soviet Socialist Republic people]]