Stefan Banach Tanja Rakovic Stefan Banach's life Stefan Banach was a Polish mathematician who was born in Krak´ow,Poland (formerly part of Austria- Hungary) 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 P lowa'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 Jagiellonian University 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 Latin, 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 Lviv, Ukraine. 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 functional analysis and made significant contributions to the theory of topo- logical vector spaces. In 1920, he wrote a dissertation that defined the \Banach space", which is a complete 2 normed vector space. Others introduced ideas related to this field such as Norbert Wiener, but Banach was the first to develop the theory. Maurice Fr´echet coined the term \Banach Space", and\Banach algebras" were also named after him. A more detailed definition of Banach space is \a real or complex normed vector space that is complete as a metric space under the metric d(x,y)= jjx − yjj 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 Banach algebra is defined to be \a Banach space where the norm satisfies jjxyjj ≤ jjxjj ∗ jjyjj". 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 geometry. The Hans-Banach theorem is named after Hans Hahn and Stefan Banach who proved it in the late 1920s. However, a man named Eduard Helly 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 Alfred Tarski and Stefan Banach publised in \Fundamenta Mathematicae." 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 axiom of choice, 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 mathematicians 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.