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Home , Eduard Helly, Hugo Steinhaus, Scottish Book, Stefan Banach

Tanja Rakovic Stefan Banach’s life

Stefan Banach was a Polish who was born in Krak´ow, (formerly part of Austria- ) on March 30, 1892. His father was a tax official named Stefan Greczek who was not married to Stefan’s mother. There is no information as to why Stefan does not have the same surname as his father, but he did have the same first name. Stefan Banach’s mother disappeared from his life after he was baptized at four days old; therefore nothing more is known about her. Even her name remains a mystery; on Banach’s birth certificate his mother is named Katarzyna Banach, but there is speculation as to whether that is her actual name. The mystery continued into Stefan Banach’s life when he asked his father about the identity of his mother, but he would not reveal anything.

Figure 1: Stefan Banach at age 44 in Lvov, 1936

Stefan Banach was later taken to his father’s birthplace in Ostrowsko, which was 50 kilometers south of Krak´ow.Stefan Greczek arranged for Banach to live with his grandmother, but she became ill and so Banach was placed with a woman named Franciszka Plowa. Banach was later introduced to a French man named Juliusz Mien. He had lived in Poland since 1870 while being a photographer and a translator of Polish literary works. Mien was the only intelligent person in Stefan Banach’s life. Therefore, it can be concluded that he was the person who encouraged him to learn more about the topics he was interested in. He recognized Banach’s talents, and allowed his to appreciate education by teaching him French. Banach demonstrated his knowledge by doing well at the international mathematics conference he attended.

Banach lived in decent conditions, especially because Franciszka Plowa’s husband was the director of Hotel Krakowski at the time Banach lived with them, and they were economically stable. Banach took this example of success, as well as his knowledge from Mien, to make himself into intellectual man he became. Even though he did not have the best childhood because his mother was not in his life, he did have a normal relationship with his father and learned much from other people.

Banach started his schooling by attending primary school, which he completed in 1902. At age 10, he enrolled in the Henryk Sienkiewicz Gymnasium No 4 in Krak´ow.He exceeded in his studies, with mathematics and the natural sciences being his best subjects. A fellow classmate named Witold Wilkosz left the school to attend a better one, but still remained in contact with Banach. The school focused more on the humanities, and was not considered as exclusive, but it had good connections to higher institutions such as the and the Polish Academy of Arts and Sciences, so Banach stayed. Nevertheless, Wilkosz and Banach were very similar in nature; both thought quickly and could solve mathematical problems quickly. Marian Albi´nskiwas also one of Banach’s classmates. He transferred to the Sobieski Grammar school because of a conflict about a failing grade between him and a teacher who taught Greek. In the school system at this time during the Austro-Hungarian annexation, a situation like this was punishable by a fine of 20 crowns (this equaled to about $9.35 at the time and $227.29 currently). Albi´nskihas described Banach as “mild mannered with a gentle sense of humor”. Banach would often tutor others for money, but he would never charge his own schoolmates. Many times, Banach and his other friend Wilkosz would discuss a challenging question. This shows Banach’s deep interest in mathematics ever since he was young. Even though the schools at the time taught , Greek and other modern languages, Banach was diligent and attended school regularly. However, he was not very interested in these classes and even his math teachers were not fully knowledgable on the subject.

The staff at the school taught rigorously overall in other subjects which allowed Banach to set himself up to higher standards. Despite Banach’s excellent grades and mathematical skills, his final examination led to poorer grades, but he still wanted to study mathematics. He felt that nothing new could be discovered in the field, so he chose to study engineering. He was an outstanding mathematician that just may have not had enough information to make the choice to stay in the field of mathematics. Later, Banach stated that his interest in mathematics was inspired by Dr. Kamil Kraft, who taught mathematics and physics. He also admitted to Professor Andrzej Turowicz that he was wrong about saying that nothing new could be discovered in mathematics because it was already so advanced. There is always something new that could be discovered or expanded upon.

After he left school, Banach’s father informed him that he was on his own, which came to no surprise because he never supported Banach much. Therefore, he left Krak´owand went to Lvov. There he enrolled in the Faculty of Engineering at Lvov Technical University. During this time, Lvov was the epicenter of Polish culture and learning. It was a diverse city, with Polish, Jewish, Ukranian, Armenian and Austrian inhabitants. Therefore, trade and knowledge of the humanities flourished. Eventually knowledge about science increased, and this was in large part due to Stefan Banach. Not much is known about his early life in Lvov, but it is likely that he continued to tutor to make a living. He continued to contribute to mathematics even throughout WWI and WWII.

He died on August 31, 1945 from lung cancer in Lvov, Ukranian S.S.R, which is now called , . He died in the Riedl family household; they were some of his closest friends. Banach died before his time at age 53, with many ideas and plans that went unfulfilled.

Stefan Banach’s mathematical works

Stefan Banach founded modern and made significant contributions to the theory of topo- logical vector spaces. In 1920, he wrote a dissertation that defined the “Banach ”, which is a complete

2 normed . Others introduced ideas related to this field such as , but Banach was the first to develop the theory. Maurice Fr´echet coined the term “”, and“Banach algebras” were also named after him. A more detailed definition of Banach space is “a real or complex that is complete as a under the metric d(x,y)= ||x − y|| induced by the norm.” This is important because the Cauchy sequences as well as the Banach spaces converge. The Banach space allows the computation of vector length and the distance between vectors, which related to a Cauchy sequence of vectors because these vectors always meet at a limit within the space.

A is defined to be “a Banach space where the norm satisfies ||xy|| ≤ ||x|| ∗ ||y||”. This means that the norm of the product is less than or equal to the product of the norms; it only works for real and complex numbers. If the Banach space becomes a normed space, then that structure is called a normed algebra. Moreover, a Banach algebra is “unital” if it has an identity element for the multiplication whose norm is 1 and “commutative” if the identity element’s multiplication is commutative.

Banach’s contribution is important because of his systematic theory of functional analysis. Before, there had only been some results that were later discovered to fit in the theory. The theory also generalized and expanded on ideas provided by Volterra, Hilbert, and Fredholm regarding integral equations.

The Hahn-Banach theorem is essential to functional analysis. It shows that the study of dual space is considered “interesting” if there are enough continuous linear functionals defined on every normed vector space. Another version of this theorem is called the Hahn-Banach separation theorem, which is commonly used in convex . The Hans-Banach theorem is named after Hans Hahn and Stefan Banach who proved it in the late 1920s. However, a man named proved that a special case exists for the space C [a,b] of continuous functions.

The Banach-Steinhaus theorem “relates the size of a certain subset of points defined relative to a family of linear mappings between topological vector spaces”. Other corollaries have stemmed from this theorem as well. For example, the uniform boundedness principle which states that any family of continuous linear operators between Banach spaces is uniformly bounded provided that it is bounded pointwise, resulted in an immediate corollary relating to the Banach-Steinhaus theorem.

The Banach Tarski paradox (which is a theorem that is so strange) first appeared in a 1926 paper by and Stefan Banach publised in “.” The paper itself was called “Sur la d´ecomposition des ensembles de points en parties respectivement congruent.” This translates to “On the decomposition of sets of points in respectively congruent parts.” The paradox states that a ball can be divided up into smaller parts that can be put back together to create two balls that are identical to the first one. The , which states that given a non-empty set, one can create a new set by choosing an element from the first set, is needed to define the breakdown. However, other question the use of this axiom because it is able to give a “non-intuitive” result. Nevertheless, this paradox contributed to the work being done on axiomatic set theory during this time.

Figure 2: An illustration of the Banach Tarski paradox.

The Banach-Alaoglu theorem states that the norm unit ball (a ball with a radius of 1) of the continuous dual X∗ of a topological vector space X is compact in the weak−∗ topology induced by the norm topology on X. This gave way to more work related to bounded sequences, convex sets, dual normed space, etc.

The Banach fixed point theorem was first stated by Banach in 1922; it is also known as the contraction mapping theorem, which is important within the field of metric spaces. It guarantees the existence of fixed points of

3 certain metric spaces and provides a unique method to find those points.

Collaboration with other scholars

In 1916, a collaboration with had a major impact on Banach’s life. Steinhaus was about to accept a post at the Jan Kazimierz University in Lvov. However he was still living in Krak´ow,and would often walk through the streets in the evenings. In one of his memoirs he wrote about one walk where he overheard two people talking about the “Lebesgue measure”. This intrigued him, so he approached the two men and introduced himself. He realized that the two men, Stefan Banach and Otto Nikodym, were mathematical geniuses. From that moment on, they would meet up regularly.

Figure 3: Jan Kazimierz University

During one meeting, Steinhaus presented to Banach a mathematical problem that he had been working on with no success. In a matter of days, Banach presented him the idea for a counterexample. This led them to work on a paper together, which they presented to a professor named Zaremba, for publication. The war delayed this publication; however it was still published in the Bulletin of the Krak´owAcademy in 1918. It was the first of Banach’s publications, called “Sur la convergence en moyenne de s´eriesde Fourier” or “On the Mean Convergence of Fourier Series.” Because of this collaboration with Steinhaus, Banach began to write even more important papers at an even quicker rate. It is possible that Banach’s interaction with Steinhaus allowed him to become more passionate about mathematics, but this cannot be said concretely. Steinhaus not only provided Banach with mathematical opportunities, but with personal ones as well. Other mathematicians began to know and recognize Banach and his works. Through these opportunities, Banach was able to excel even more in his field of study. Through Hugo Steinhaus, Banach also met his wife, Lucja Braus. They were married in 1920 in Krak´ow,and later in 1922 they had a son named Stefan Banach Jr.

During this time, a man named Zygmunt Janiszewski was one of the Polish intellectuals dedicated to making mathematics even better by creating a journal dedicated to mathematics, which was connected to set theory. He founded the first volume of the periodical “Fundamenta Mathematicae”, which contained Banach’s published paper, called “On the Functional Equation.” This was a notable achievement because this was the volume in which Banach’s publication was in the new periodical for the first time.

4 The team also set up the Mathematical Societ of Krak´owin 1919. Professor Zaremba was elected as the first President of the Society, which gave it even more quality. Banach lectured to the society twice and continued to produce excellent papers. He constantly contributed greatly to the field of mathematics, which allowed him to qualify for an assistantship to Professor Antoni Lomnicki at Lvov Technical University in 1920. There he lectured in mathematics and was set on receiving his doctorate. Therefore, he submitted a dissertation under the supervision of ProfessorLomnicki, which was not the traditional way to a doctorate, but Banach had no mathematics qualifications from a university. He did have, however, a brilliant mind and superior colleagues that knew the quality of his work. Thus, an exception was made for him and he was allowed to submit his paper On Operators Defined on Abstract Sets and their Applications to Integral Equations. His thesis “is sometimes said to mark the birth of functional analysis.” In order to confirm his PhD candidacy, his evaluators compiled his work. They discovered that Banach’s work was outstanding PhD material, but they still had to externally assess him. One day, Banach was called in to the Dean’s office of Jan Kazimierz University to answer questions and prove that he was worthy of his PhD status. He was completely unaware that a commission from Warsaw came just to evaluate him. He also presented a thesis about measure theory that allowed the university of Jan Kazimierz in Lvov to award him his habilitation. Today, it would likely not be possible to obtain a PhD in this way.

In 1922 Banach became a professor at the university, and two years later he was elected as a “Corresponding Member of the Polish Academy of Arts and Sciences” In 1924, Banach became a full professor and spent this time in Paris to lecture and contribute to mathematical work there as well. Still, one of his goals was to continue publishing mathematical works. He did so by writing papers and texts for high schools. Eventually, Banach became on of the greatest experts in functional analysis, so he created the Lvov School of Mathematics, along with his colleague Steinhaus. In 1929, the duo started a journal named , and they were the first editors.

Following the publication of Banach’s book called “Theory of Linear Operations” in 1931, the world finally began to recognize Banach’s work. It was the first volume of a series of ”Mathematical Monographs”. This was the foundation of the first textbook about functional analysis.

In 1932, Banach became the vice president of the Polish Mathematical Society. Even though he had new, more demanding responsibilities, he took them on without a problem. The most important part of his work were the publications. Banach’s contribution to the “Mathematical Monographs” (his work about the theory of linear operations) allowed more mathematicians to write about various other topics that were equally as important. Therefore, in a short time the “Mathematical Monographs” became on of the most important scientific periodicals in history.

Moreover, Banach wrote textbooks regarding advanced mathematics; this worked well with his lectures. He wrote volumes I and II of “Differential and Integral Calculus” in 1929 and 1930, respectively. Additionally, he wrote volumes I and II of “Mechanics - In the Scope of Academic Studies” in 1938. Banach worked with Sto˙zekand Sierpi´nskito write grammar school textbooks, although these were published while Banach was undergoing difficult and dramatic circumstances. Regarding Banach’s conditions, Steinhaus wrote that Banach was able to work under any conditions whether they were difficult or not. He liked coffee-houses, and began to disregard his daily schedule which led him into debt. This is why he attempted to write textbooks to save himself. During this time, Banach received help from Professor Benedykt Fuli´nski,who guaranteed to pay back Banach’s debts. While Fuli´nskipromised to do this, he also wanted Banach to learn how to be economically stable for himself. Therefore, Fuli´nskitold Banach to set aside some income every month, which would allow him to change his spending habits. Only Banach’s income from his textbooks allowed him to pay off his debts little by little, but when he received a prize from the Polish Academy of Arts and Sciences in 1939, then he was able to fully pay off his debts.

Mathematicians all over the world were intrigued by Banach’s work. At the 1936 International Congress of

5 Mathematics in Oslo, Banach gave a lecture on “The Theory of Operations and its Significance in Analysis.” He also praised the Lvov University and spoke about plans to develop their work further.

Stefan Banach also worked with Kuratowski at Lvov Technical University. They worked on a few joint papers during this time.

Figure 4: This is the cafe where the mathematicians regularly met up at.

The environment in which Banach did his work was different; he liked going to caf´esin Lvov to do mathematical work with his colleagues and on his own there. He liked the background noise and the music, which did not prevent him from concentrating on his work. In general, the Lvov School of Mathematics frequently held meetings at the Scottish Caf´e. It was located near the university and was an integral part of the mathematicians’ scientific work, as well as their contributions to society. Stefan Banach, , and Stanislaw Mazur created the most efficient team there. A reason why the team visited the caf´eso frequently was because it had marble table tops which could easily be written on and erased. Hugo Steinhaus was once invited to a meeting (being invited to one of these meetings was said to be “tantamount to being knighted” and he said that one meeting lasted 17 hours, resulting in the proof of a postulate regarding Banach spaces. It was written on the tabletops, but no permanent record of it was made because it was probably wiped away by an employee. Similarly, many other proofs by Banach and his students were erased. There is no information to how the mathematicians reacted to their work being erased. It is known however that at these meetings there was always something exciting and new to talk about, which made it possible to have an intellectual conversation concerning mathematics. Stanislaw Ulam recalled that these sessions produced collaboration and concentration with such a high level of intensity that he has not seen before. Eventually, Banach’s wife Lucia bought a notebook (later named the “” that she gave to the employees of the Scottish Caf´e.She told them to give it to any mathematician who wanted to use it. Therefore, in just a few years the notebook contained permanent records of challenging mathematical problems with some solutions. Numerous mathematicians contributed to the notebook, and anyone who entered a problem would offer a prize (including alcohol, caviar, bacon, and a live goose) to the next mathematician who could pose a solution. The Scottish Book experienced some troubling times as well, because when the Soviets arrived in Lvov after the oubreak of

6 WWII, there were entries in the book that indicated Soviet mathematicians made them. They also promised prizes to whoever could solve their problems.

The “Scottish Book” was basically a “holy relic”. It was extremely important to other mathematicians all over the world, so much that it was transcribed in type and copies were distributed among them.

Historical events that marked Stefan Banach’s life.

Figure 5: This was the city of Lvov before WWI.

Lvov was under Austrian control at the time Banach studied there. Even while he was living in Poland during his youth, it was controlled by Russia. In a sense, Poland did not even exist because it was dominated by everything Russian, including the sole Russian language university in Warsaw. The outbreak of World War One in the summer of 1914 led to Banach leaving Lvov for Krak´owa short time after graduation. The Russian troops occupied the city, and Banach was not able to serve in the army because the vision in his left eye had deteriorated. However, he contributed in another way by building roads and by teaching at local schools in Krak´ow.Despite the war, he still was determined to keep learning about mathematics, and attended lectures at the Jagiellonian University in Krak´ow.He was still passionately involved with mathematics, and continued to learn more through discussions he had with Otto Nikodym and Witold Wilkosz.

A second war was now beginning to break out. The Soviet army began to occupy Lvov on September 22, 1939. However, schools were still open and continued to be taught in Polish. Banach was still a professor and dean of the Mathematics and Philosophy Faculty at Lvov University. Then he was elected as a Corresponding Member of the Ukrainian Soviet Republic’s Academy of Sciences. He reluctantly accepted the nomination to be a delegate member of the Lvov City Council because he was reluctant about getting involved in politics. Even with his dedication to the city of Lvov, he constantly kept up with the work and achievements of his former coworkers and students in the USA.

Stefan Banach Jr. later said that his father was invited to a conference in Kiev, which was scheduled two days before the war broke out between Germany and the . He went, and made it just in time before

7 Figure 6: This was the city of Lvov during WWI. There is a significant amount of troops pictured here showing how influential the war was. the war started. He took the last train to Lvov and arrived right before the Germans took over. Banach Jr. asked his father why he just did not stay in Kiev, to which Banach responded that he loved his family and that was the way every Banach behaved.

German troops entered Lvov on the night of June 30, 1941- July 1, 1941 which was three days after the Soviets fled the city. The citizens who remained were still shocked due to the crimes of the NKGB (Soviet Secret Police) on prisoners. From July 2-4 1941, the Germans arrested 23 professors from the Jan Kazimierz University, Lvov Polytechnic and the Veterinary Academy. The professors were all shot on the Wulka Hills near Lvov on July 4, 1941 at dawn. Others who were spared, later died from natural causes as well as deprivation of food, shelter, water, etc. caused by the war. Banach was lucky to have survived this mass murder.

The Germans continued to occupy Lvov until 1944, and during this time Banach along with other scholars and students were able to gain employment at the Institute for Studies. It was directed by Professor , and the purpose of it was to conduct experiments that required the feeding of lice with human blood. This was very important to the German military, so whoever worked there received a document that protected them from German persecution. At the outbreak of WWII, the Biology Faculty at the Jan Kazimierz University in Lvov produced large amounts of the vaccine against epidemic typhus, since the Polish Government’s Ministry of the Army required it. A new institute of Bacteriology and Sanitary science was created to further research opportunities. Typhus vaccines were still being produced in large quantities, but this time most of them were being sent to the Soviet Union to protect the . The German armed forces attacked the Soviet Union, entered Lvov, and then renamed the institute to “Institut f¨urFleckfieber und Virusforschung des OKH, which was under German control. Professor Weigl was left as director of the institute, and required to increase the production of the vaccines. The institute was too small to continue the production, so it overtook a building that used to be a grammar school and expanded. The entire production of the vaccine was designated for use by the German land armies.

At first, the scholars faced an extremely difficult problem because they were unemployed. This is why Weigl continued to run the institute. He wanted to give them an opportunity to continue to work and produce

8 results. He trusted them and so the Institute grew. A unique group was formed including academics, young people who had conspiracies against the occupiers, and fighters in the underground resistance. The biggest reason that they wanted to work there was because upon employment, each person received a work permit that would protect them from the Germans. Therefore, if it was not for Weigl, there is a large possibility that most if not all the employees at the Institute for Typhus Studies would be killed by the Germans.

Banach’s work at the Institute lasted until the Nazis stopped occupying Lvov in July 1944. Right after, he was offered a position as the Chair of Mathematics at the Jagiellonian University in Krak´ow. He was more comfortable working there than as a lice feeder. However, a serious illness and eventual death prevented this.

Years later, Stefan Banach Jr. recalled that his father visited him in Krak´ow.Stefan Banach seemed to look better, and his sense of hope was reignited. He wanted to switch from studying mathematical problems to studying physics problems that might win him the Nobel Prize. Wladys law Nikliborc took care of Banach during the final months of his life. Banach Jr. has said how he did not know how such a person could have so much “heart and courage.”

Significant historical events around the world during Stefan Banach’s life

The most significant events that happened during Stefan Banach’s life were WWI and WWII. During WWI, Austria declared war on Serbia, Germany declared war on Russia and , and Britain declared war on Germany. This set up unprecedented levels of destruction, genocide, and the development of new technology. WWI did not influence Banach as much as WWII did, but it is still important to note that Banach continued to work on his mathematical work, especially because he could not fulfill his military service. The wars certainly had a great influence on his life, and possibly the extent of his work. Despite all the destruction happening around him and the world, he continued to work and contribute to the mathematics field by writing textbooks and papers, attending lectures, and conducting research.

Other events happening around Banach’s time included the Russian Revolution in 1917, which was a violent upheaval in Russia that overthrew the czarist government. At the same time, the US declared war on Austria- Hungary and an armistice between the newly created Russian Bolshevik government and the Germans was signed. In 1918, the Russian Civil War between the Bolsheviks and Anti-Bolsheviks broke out, with the Bolsheviks winning and keeping power. In 1920, the U.S. Department of Justice led a “red hunt” that deported thousands of aliens. Then, Lenin died in 1924, so Stalin took over and implemented his demanding policies on the USSR.

In 1927, the German economy collapsed. Later however, the Nazis gained much support in the elections and Hitler became the German chancellor. This is the point where the Nazi terror began. The Germans were occupying the Rhineland in 1936, and at the same time a war between China and Japan began, which continued through WWII. The Germans continued building power by invading nearby countries (i.e. Poland). In 1939 WWII began, and Germany attacked the Balkans and Russia. The US got involved in WWII when its fleet was attacked by the Japanese at Pearl Harbor. To stop this massive war, the Yalta Conference (with leaders Roosevelt, Churchill, and Stalin) discussed plans to defeat Germany once and for all. Then Roosevelt died, and Hitler committed suicide, so his reign of terror was over. However, the US became violent by dropping atomic bombs on Hiroshima and Nagasaki in Japan. On a lighter note, the first electronic computer, called ENIAC was built. Throughout this time, there was much violence and destruction, but there was also an unprecedented, significant advancement in technology. In 1946, Winston Churchill warned of Soviet expansion in his “Iron Curtain” speech, while the Soviets become the ones gaining power instead of Germany.

All of this violence did not extremely affect Banach, although it did hinder the production of some of his work. He obviously could not work on mathematics when he was a lice feeder. During this time, one could not help

9 but wonder what else Banach could have achieved had he not been interrupted by the Germans.

Figure 7: This is a map of Poland during WWI.

Significant mathematical progress during the Stefan Banach’s lifetime

An important publishing project that Banach was pursuing was a series of Mathematical Monographs. This was made possible due to the editing skills of Banach and Hugo Steinhaus, as well as a few professors from Warsaw. The first volume in this series was written by Banach, and he called it the Th´eoriedes Op´erations lin´eaires (Theory of Linear Operations). This was originally written in Polish, but this French version quickly became accredited with many people, perhaps because a substantially larger amount of people knew French better than Polish. Even more people took him seriously, which allowed him to give an address at the International Congress of Mathematicians in Oslo. He described the work and dedication shown by the school in Lvov that he attented, as well as plans to develop their mathematical ideas further.

The main focus of Banach’s work was functional analysis, which brought him fame but also significant contri- butions and changes to the mathematics field. His other contributions included the theory of real functions, measure theory, integration, the theory of sets, and orthogonal series.

Functional analysis continued to spread among mathematics centers all over the world. His ideas were the to the ideas of other mathematicians. Banach was known as the pillar, architect, and founder of the Polish School of Mathematics.

Connections between history and the development of mathematics

Right before the beginning of WWII, Banach was elected as president of the Polish Mathematical Society. When the war started, Soviet troops occupied Lvov. However, this did not affect Banach in a negative way because he had good connections to Soviet mathematicians. He visited them often and they treated him well.

10 They allowed him to hold his chair at the Faculty of Science at the university in Lvov and eventually, he became the Dean of the university. Despite another war, Banach continued to research, write, lecture and go to cafes. He also attended conferences in the Soviet Union; a reason for this may be to keep up his good relationship with the Soviets. Banach was in Kiev but returned immediately to Lvov when Germany invaded the Soviet Union.

Due to the Nazi occupation of Lvov in 1941, Banach lived under difficult and harsh conditions. He was suspected of illegally dealing German currency, which landed him in prison, but he was released shortly after. Possibly the most influential and tragic event to happen was a period in time when many Polish scholars were murdered. This directly affected Banach because his doctoral supervisor, Lomnicki, died. Banach then ended up working in a German institute as a feeder of lice. His job was to feed blood to lice, which was the only way at the time to create a vaccine against the typhus disease. Considering his education and the level of his intelligence, this job would be unacceptable for someone like him in the modern world, but at the time the war had such a large impact that there were many unusual jobs that needed to be filled, regardless of education level. It was hard Banach to do something related to mathematics because the whole world’s attention was on the war, so he had to focus on it as well. This distracted him from focusing on his work, especially because he had the same job until 1944. When the Soviets regained control of Lvov, Banach met up with some people he knew, including Sobolev. By this time, he had noticed that Banach was very ill. Sobolev recalled, that despite the years of war that Banach went through as well as the serious diease he was fighting, he was still “the same sociable, cheerful, and extraordinary well-meaning and charming Stefan Banach whom [he] had seen in Lvov before the war”. That is how Banach remained in the memories of the people who new him, despite his illness. Eventually, Banach planned to go to Krak´owafter the war to become the chair of mathematics at the Jagiellonian University. However, he died in 1945 in Lvov due to lung cancer.

Remarks

Stefan Banach’s life is similar to Alan Turing’s life in that they both did not serve in the army when the wars were happening during their time period. However, they both contributed in a significant way mathematically, Turing by creating a machine that decoded “Enigma”, or the codes created by the Germans, while Banach continued to teach and write ground breaking papers about mathematics that furthered the field.

In 1946, the Polish Mathematical Society established the Stefan Banach Award, while the Polish Academy of Sciences established a special Stefan Banach medal 100 years after his birth. This medal is awarded to someone who demonstrates outstanding achievements related to mathematics.

Many schools and streets continue to be named after Banach. For example, in 1972 the International Stefan Banach Mathematical Center was created within the Institute of Mathematics: Polish Academy of Sciences. The Center hosts conferences with bright young mathematicians and specialists from all over the world, which has been useful for international cooperation within mathematics.

In 1999, a statue of Stefan Banach was unveiled in Krak´owon the 54th anniversary of his death. He was also featured on Polish stamps in 1982.

At a memorial conference for Banach, Hugo Steinhaus gave an important summary of Banach’s life, and stated that Banach contributed the most to not only Polish mathematics, but mathematics all over the world, more than anyone else. He had the talent, the “spark of genius”, the persistence, and the determination to work is what made him one of the greatest mathematicians of all time.

References

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6. http://l7.alamy.com/zooms/b6bf846938a14482a96b89e13763f303/austrian-troops-in-lviv-1915-e0048r. jpg

7. http://kielich.amu.edu.pl/Stefan_Banach/e-biography.html

8. https://www.impan.pl/en/insitute/history

9. http://mathworld.wolfram.com/Banach-SteinhausTheorem.html

10. http://mathworld.wolfram.com/Banach-AlaogluTheorem.html

11.http://kielich.amu.edu.pl/Stefan_Banach/jpg/uniwersytet2x.jpg

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