Stefan Banach
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Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a
Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern
functional analysis 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, and an original member of the
Lwów School of Mathematics The Lwów school of mathematics ( pl, lwowska szkoła matematyczna) was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to ...
. 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. Born in
Kraków Kraków (), or Cracow, is the second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city dates back to the seventh century. Kraków was the official capital of Poland until 1596 ...
to a family of
Goral The gorals are four species in the genus ''Naemorhedus''. They are small ungulates with a goat-like or antelope-like appearance. Until recently, this genus also contained the serow species (now in genus '' Capricornis''). Etymology The original ...
descent, Banach showed a keen interest in mathematics and engaged in solving mathematical problems during school recess. After completing his secondary education, he befriended
Hugo Steinhaus Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) was a 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 Un ...
, with whom he established the
Polish Mathematical Society The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Un ...
in 1919 and later published the scientific journal '' Studia Mathematica''. In 1920, he received an assistantship at the
Lwów Polytechnic Lviv Polytechnic National University ( ua, Націона́льний університе́т «Льві́вська політе́хніка») is the largest scientific university in Lviv, Ukraine. Since its foundation in 1816, it has bee ...
, subsequently becoming a professor in 1922 and a member of the
Polish Academy of Learning The Polish Academy of Arts and Sciences or Polish Academy of Learning ( pl, Polska Akademia Umiejętności), headquartered in Kraków and founded in 1872, is one of two institutions in contemporary Poland having the nature of an academy of scie ...
in 1924. Banach was also a co-founder of the
Lwów School of Mathematics The Lwów school of mathematics ( pl, lwowska szkoła matematyczna) was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to ...
, a
school of thought A school of thought, or intellectual tradition, is the perspective of a group of people who share common characteristics of opinion or outlook of a philosophy, discipline, belief, social movement, economics, cultural movement, or art movement. ...
comprising some of the most renowned Polish mathematicians of the
interwar period In the history of the 20th century, the interwar period lasted from 11 November 1918 to 1 September 1939 (20 years, 9 months, 21 days), the end of the World War I, First World War to the beginning of the World War II, Second World War. The in ...
(1918–1939). Some of the notable
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by s ...
s that bear Banach's name include
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s,
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s,
Banach measure In the mathematics, mathematical discipline of measure theory, a Banach measure is a certain type of finite measure, content used to formalize geometric area in problems vulnerable to the axiom of choice. Traditionally, intuitive notions of are ...
s, the
Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
, the
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
, the
Banach–Steinhaus theorem In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
, the
Banach–Mazur game In general topology, set theory and game theory, a Banach– Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire s ...
, the
Banach–Alaoglu theorem In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common ...
, and the
Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
.


Life


Early life

Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in
Kraków Kraków (), or Cracow, is the second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city dates back to the seventh century. Kraków was the official capital of Poland until 1596 ...
, then part of the
Austro-Hungarian Empire Austria-Hungary, often referred to as the Austro-Hungarian Empire,, the Dual Monarchy, or Austria, was a constitutional monarchy and great power in Central Europe between 1867 and 1918. It was formed with the Austro-Hungarian Compromise of ...
, into a
Góral Góral is a Polish habitational surname A toponymic surname or topographic surname is a surname derived from a place name.
Roman Catholic Roman or Romans most often refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of ancient Rome *'' Epistle to the Romans'', shortened to ''Romans'', a lette ...
family and was subsequently baptised by his father. Banach's parents were Stefan Greczek and Katarzyna Banach, both natives of the
Podhale Podhale (literally "below the mountain pastures") is Poland's southernmost region, sometimes referred to as the "Polish Highlands". The Podhale is located in the foothills of the Tatra range of the Carpathian mountains. It is the most famous ...
region. Greczek was a soldier in the
Austro-Hungarian Army The Austro-Hungarian Army (, literally "Ground Forces of the Austro-Hungarians"; , literally "Imperial and Royal Army") was the ground force of the Austro-Hungarian Dual Monarchy from 1867 to 1918. It was composed of three parts: the joint arm ...
stationed in Kraków. Little is known about Banach's mother. According to his baptismal certificate, she was born in Borówna and worked as a domestic helper. 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
marry Marriage, also called matrimony or wedlock, is a culturally and often legally recognized union between people called spouses. It establishes rights and obligations between them, as well as between them and their children, and between t ...
; he was a
private Private or privates may refer to: Music * " In Private", by Dusty Springfield from the 1990 album ''Reputation'' * Private (band), a Denmark-based band * "Private" (Ryōko Hirosue song), from the 1999 album ''Private'', written and also recorde ...
and as the mother was too poor to support the child, the couple decided that he should be reared by family and friends. 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. 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. In 1902, Banach, aged 10, enrolled in Kraków's ''IV Gymnasium'' (also known as the '' Goetz Gymnasium''). While the school specialized in the
humanities Humanities are academic disciplines that study aspects of human society and culture. In the Renaissance, the term contrasted with divinity and referred to what is now called classics, the main area of secular study in universities at the t ...
, 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. Later in life Banach credited Dr. Kamil Kraft, the mathematics and physics teacher at the school, with kindling his interests in mathematics. 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. After obtaining his ''
matura or its translated terms (''Mature'', ''Matur'', , , , , , ) is a Latin name for the secondary school exit exam or "maturity diploma" in various European countries, including Albania, Austria, Bosnia and Herzegovina, Bulgaria, Croatia, Czech ...
'' (high school degree) at age 18 in 1910, Banach moved to Lwów (today called
Lviv Lviv ( uk, Львів) is the largest city in western Ukraine, and the seventh-largest in Ukraine, with a population of . It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main cultural centres of Ukraine ...
) with the intention of studying at the
Lwów Polytechnic Lviv Polytechnic National University ( ua, Націона́льний університе́т «Льві́вська політе́хніка») is the largest scientific university in Lviv, Ukraine. Since its foundation in 1816, it has bee ...
. 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. 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. When
World War I World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
broke out, Banach was excused from military service due to his left-handedness and poor vision. When the
Russian Army The Russian Ground Forces (russian: Сухопутные войска ВSukhoputnyye voyska V}), also known as the Russian Army (, ), are the land forces of the Russian Armed Forces. The primary responsibilities of the Russian Ground Force ...
opened its offensive toward Lwów, Banach left for Kraków, where he spent the rest of the war. He made his living as a
tutor TUTOR, also known as PLATO Author Language, is a programming language developed for use on the PLATO system at the University of Illinois at Urbana-Champaign beginning in roughly 1965. TUTOR was initially designed by Paul Tenczar for use in co ...
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 and Kazimierz Żorawski, but little is known of that period of his life.


Discovery by Steinhaus

In 1916, in Kraków's '' Planty'' park, Banach encountered Professor
Hugo Steinhaus Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) was a 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 Un ...
, 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 In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
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 Otto Marcin Nikodym (3 August 1887 – 4 May 1974) (also Otton Martin Nikodým) was a Polish mathematician. Education and career Nikodym studied mathematics at the University of Jan Kazimierz (UJK) in Lvov (today's University of Lviv). Imm ...
. 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 Władysław Ślebodziński () (February 6, 1884 – January 3, 1972) was a Polish mathematician. Władysław Ślebodziński was born in Pysznica, Poland and educated at the Jagiellonian University in Kraków (1903-1908) where he subsequently he ...
,
Leon Chwistek Leon Chwistek (Kraków, Austria-Hungary, 13 June 1884 – Barvikha near Moscow, Russia, 20 August 1944) was a Polish avant-garde painter, theoretician of modern art, literary critic, logician, philosopher and mathematician. Career and philosophy ...
, Alfred Rosenblatt and
Włodzimierz Stożek Włodzimierz Stożek (23 July 1883 – 3 or 4 July 1941) was a Polish mathematician of the Lwów School of Mathematics. Head of the Mathematics Faculty on the Lwów University of Technology. He was arrested and murdered together with his two son ...
) also established a mathematical society, which eventually became the
Polish Mathematical Society The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Un ...
. The society was officially founded on 2 April 1919. It was also through Steinhaus that Banach met his future wife, Łucja Braus.


Interbellum

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 A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''l ...
without actually graduating from a university. The doctoral thesis, accepted by King John II Casimir University of Lwów in 1920 and published in 1922, 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. 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 Antoni Marian Łomnicki (17 January 1881 – 4 July 1941) was a Polish mathematician. Antoni Łomnicki was educated at Jan Kazimierz University in Lwów in Poland and the University of Göttingen in Germany. In 1920 he became professor of the L ...
, in 1927, Banach received his own chair. In 1924 he was accepted as a member of the
Polish Academy of Learning The Polish Academy of Arts and Sciences or Polish Academy of Learning ( pl, Polska Akademia Umiejętności), headquartered in Kraków and founded in 1872, is one of two institutions in contemporary Poland having the nature of an academy of scie ...
. At the same time, from 1922, Banach also headed the second Chair of Mathematics at
University of Lwów The University of Lviv ( uk, Львівський університет, Lvivskyi universytet; pl, Uniwersytet Lwowski; german: Universität Lemberg, briefly known as the ''Theresianum'' in the early 19th century), presently the Ivan Franko Na ...
. Young and talented, Banach gathered around him a large group of mathematicians. The group, meeting in the
Scottish Café The Scottish Café ( pl, Kawiarnia Szkocka) was a café in Lwów, Poland (now Lviv, Ukraine) where, in the 1930s and 1940s, mathematicians from the Lwów School of Mathematics collaboratively discussed research problems, particularly in functio ...
, 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 Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
-
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. First published in Polish in 1931, the next 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. In 17 June 1924, Banach become a correspondence member of the Polish Academy of Sciences and Fine Arts in Kraków.


World War II

After the
invasion of Poland The invasion of Poland (1 September – 6 October 1939) was a joint attack on the Republic of Poland by Nazi Germany and the Soviet Union which marked the beginning of World War II. The German invasion began on 1 September 1939, one week aft ...
by
Nazi Germany Nazi Germany (lit. "National Socialist State"), ' (lit. "Nazi State") for short; also ' (lit. "National Socialist Germany") (officially known as the German Reich from 1933 until 1943, and the Greater German Reich from 1943 to 1945) was ...
and the
Soviet Union The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national ...
, Lwów came under the control of the
Soviet Union The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national ...
for almost two years. Banach, from 1939 a corresponding member of the
Academy of Sciences of Ukraine The National Academy of Sciences of Ukraine (NASU; uk, Національна академія наук України, ''Natsional’na akademiya nauk Ukrayiny'', abbr: NAN Ukraine) is a self-governing state-funded organization in Ukraine th ...
, and on good terms with
Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
mathematicians, had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities. After the German takeover of Lwów in 1941 during
Operation Barbarossa Operation Barbarossa (german: link=no, Unternehmen Barbarossa; ) was the invasion of the Soviet Union by Nazi Germany and many of its Axis allies, starting on Sunday, 22 June 1941, during the Second World War. The operation, code-named after ...
, all universities were closed and Banach, along with many colleagues and his son, was employed as a lice feeder at Professor
Rudolf Weigl Rudolf Stefan Jan Weigl (2 September 1883 – 11 August 1957) was a Polish biologist, physician and inventor, known for creating the first effective vaccine A vaccine is a biological preparation that provides active acquired immunity to a ...
's
Typhus Typhus, also known as typhus fever, is a group of infectious diseases that include epidemic typhus, scrub typhus, and murine typhus. Common symptoms include fever, headache, and a rash. Typically these begin one to two weeks after exposure. ...
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 From 1933 to 1945, Nazi Germany operated more than a thousand concentration camps, (officially) or (more commonly). The Nazi concentration camps are distinguished from other types of Nazi camps such as forced-labor camps, as well as concen ...
. After the Soviet
Red Army The Workers' and Peasants' Red Army (Russian: Рабо́че-крестья́нская Кра́сная армия),) often shortened to the Red Army, was the army and air force of the Russian Soviet Federative Socialist Republic and, after ...
recaptured
Lviv Lviv ( uk, Львів) is the largest city in western Ukraine, and the seventh-largest in Ukraine, with a population of . It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main cultural centres of Ukraine ...
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 deporting Poles from annexed formerly Polish eastern territories, Banach began preparing to leave the city and settle in
Kraków Kraków (), or Cracow, is the second-largest and one of the oldest cities in Poland. Situated on the Vistula River in Lesser Poland Voivodeship, the city dates back to the seventh century. Kraków was the official capital of Poland until 1596 ...
, Poland, 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, he was diagnosed with
lung cancer Lung cancer, also known as lung carcinoma (since about 98–99% of all lung cancers are carcinomas), is a malignant lung tumor characterized by uncontrolled cell growth in tissue (biology), tissues of the lung. Lung carcinomas derive from tran ...
and was permitted to stay in Lwów. He died on 31 August 1945, aged 53. His funeral at the
Lychakiv Cemetery Lychakiv Cemetery ( uk, Личаківський цвинтар, translit=Lychakivs’kyi tsvyntar; pl, Cmentarz Łyczakowski we Lwowie), officially State History and Culture Museum-Preserve "Lychakiv Cemetery" ( uk, Державний істор ...
(''Cmentarz Łyczakowski'') was attended by hundreds of people.


Contributions

Banach's dissertation, completed in 1920 and published in 1922, formally axiomatized the concept of a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
and laid the foundations for the area of
functional analysis 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. 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 An eponym is a person, a place, or a thing after whom or which someone or something is, or is believed to be, named. The adjectives which are derived from the word eponym include ''eponymous'' and ''eponymic''. Usage of the word The term ''epon ...
ous naming of these spaces after him. The theory of what came to be known as
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s had antecedents in the work of the Hungarian mathematician
Frigyes Riesz Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...
(published in 1916) and contemporaneous contributions from Hans Hahn and Norbert Wiener. 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''. Likewise, Banach's fixed point theorem, based on earlier methods developed by
Charles Émile Picard Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was ...
, was included in his dissertation, and was later extended by his students (for example in the
Banach–Schauder theorem In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a bounded or continuous linear opera ...
) and other mathematicians (in particular
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Bro ...
and Poincaré and Birkhoff). The theorem did not require linearity of the
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
, and applied to any complete
Cauchy space In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool deriv ...
(in particular to any complete metric space). The
Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
is one of the fundamental theorems of functional analysis. Further theorems related to Banach are: *
Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
*
Banach–Steinhaus theorem In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerst ...
*
Banach–Alaoglu theorem In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common ...
*
Banach–Stone theorem 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 brief, the Banach–Stone theorem allows one to recove ...


Recognition

In 1946, the
Stefan Banach Prize The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Un ...
(Polish: ''Nagroda im. Stefana Banacha'') was established by the
Polish Mathematical Society The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Un ...
. In 1992, the
Institute of Mathematics of the Polish Academy of Sciences The Institute of Mathematics of the Polish Academy of Sciences is a research institute of the Polish Academy of Sciences.Stefan Banach Medal Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
for outstanding achievements in mathematical sciences. Since 2009, the
International Stefan Banach Prize The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Un ...
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". Stefan Banach is the patron of a number of schools and streets including in
Warsaw Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland, and its population is officia ...
,
Lviv Lviv ( uk, Львів) is the largest city in western Ukraine, and the seventh-largest in Ukraine, with a population of . It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main cultural centres of Ukraine ...
,
Świdnica Świdnica (; german: Schweidnitz; cs, Svídnice; szl, Świdńica) is a city in south-western Poland in the region of Silesia. As of 2019, it has a population of 57,014 inhabitants. It lies in Lower Silesian Voivodeship, being the seventh large ...
,
Toruń )'' , image_skyline = , image_caption = , image_flag = POL Toruń flag.svg , image_shield = POL Toruń COA.svg , nickname = City of Angels, Gingerbread city, Copernicus Town , pushpin_map = Kuyavian-Pom ...
and
Jarosław Jarosław (; uk, Ярослав, Yaroslav, ; yi, יאַרעסלאָוו, Yareslov; german: Jaroslau) is a town in south-eastern Poland, with 38,970 inhabitants, as of 30 June 2014. Situated in the Subcarpathian Voivodeship (since 1999), previ ...
. In 2001, a minor planet 16856 Banach, discovered by
Paul Comba Paul G. Comba (1926 – April 5, 2017) was an Italian-American computer scientist, an amateur astronomer and a prolific discoverer of minor planets This is a list of minor-planet discoverers credited by the Minor Planet Center with the disco ...
in 1997, was named after him. In 2012, the
National Bank of Poland The Narodowy Bank Polski (; the National Bank of Poland), often abbreviated to NBP, is the central bank of Poland, founded in 1945. It controls the issuing of Poland's currency, the Polish złoty. The bank is headquartered in Warsaw, and has bra ...
celebrated the mathematician's achievements by issuing a series of commemorative coins designed by Robert Kotowicz (golden 200- zloty coin, silver 10-zloty coin and Nordic Gold 2-zloty coin). In 2016, a commemorative bench featuring Banach and
Otto Nikodym Otto Marcin Nikodym (3 August 1887 – 4 May 1974) (also Otton Martin Nikodým) was a Polish mathematician. Education and career Nikodym studied mathematics at the University of Jan Kazimierz (UJK) in Lvov (today's University of Lviv). Imm ...
was unveiled in Kraków's