Search Results for Banach

Biographies

1. Banach biography
   
   • Stefan Banach .
     
   • Stefan Banach's father was Stefan Greczek.
     
   • The first thing to notice is that Banach was not his father's surname, but Banach was given his father's first name.
     
   • Stefan Greczek was a tax official who was not married to Banach's mother who vanished from the scene after Stefan was baptised, when he was only four days old, and nothing more is known of her.
     
   • The name given as Stefan's mother on his birth certificate is Katarzyna Banach.
     
   • In later life Banach tried to find out who his mother was but his father refused to say anything except that he had been sworn to secrecy over her identity.
     
   • It was to Ostrowsko, to his grandmother's home, that Banach was taken after his baptism.
     
   • However, when Banach's grandmother took ill, Stefan Greczek arranged for his son to be brought up by Franciszka Plowa who lived in Krakow with her daughter Maria.
     
   • Although Banach never went back to live with his grandmother, he did visit her frequently as he grew up.
     
   • Maria's guardian was a French intellectual Juliusz Mien and he quickly recognised the talents that Banach had.
     
   • Banach attended primary school in Krakow, leaving the school in 1902 to begin his secondary education at the Henryk Sienkiewicz Gymnasium No 4 in Krakow.
     
   • By a fortunate coincidence, one of the students in Banach's class was Witold Wilkosz who himself went on to become a professor of mathematics.
     
   • Banach, however, remained at Henryk Sienkiewicz Gymnasium No 4 although he maintained contact with Wilkosz.
     
   • During his first few years at the Gymnasium Banach achieved first class grades with mathematics and natural sciences being his best subjects.
     
   • A fellow school pupil recalled Banach at this period in his life (see [The life of Stefan Banach (Boston, 1996).',3)">3]):- .
     
   • [Banach] was pleasant in dealings with his colleagues, but outside of mathematics he was not interested in anything.
     
   • Also, while Banach was faster in mathematical problems, Wilkosz was phenomenally fast in solving problems in physics, which were of no interest to Banach.
     
   • On leaving school Banach and Wilkosz both wanted to study mathematics, but both felt that nothing new could be discovered in mathematics so each chose to work in a subject other than mathematics.
     
   • Banach chose to study engineering, Wilkosz chose oriental languages.
     
   • Banach's father had never given his son much support, but now once he left school he quite openly told Banach that he was now on his own.
     
   • Banach left Krakow and went to Lvov where he enrolled in the Faculty of Engineering at Lvov Technical University.
     
   • It is almost certain that Banach, without any financial support, had to support himself by tutoring.
     
   • It is not entirely clear what Banach's plans were in 1914 but the outbreak of World War I in August, shortly after his graduation, saw Banach leave Lvov.
     
   • Lvov was, at the time Banach studied there, under Austrian control as it had been from the partition of Poland in 1772.
     
   • In Banach's youth Poland, in some sense, did not exist and Russia controlled much of the country.
     
   • Banach was not physically fit for army service, having poor vision in his left eye.
     
   • A chance event occurred in the spring of 1916 which was to have a major impact on Banach's life.
     
   • The youngsters were Stefan Banach and Otto Nikodym.
     
   • Steinhaus told Banach of a problem which he was working on without success.
     
   • After a few days Banach had the main idea for the required counterexample and Steinhaus and Banach wrote a joint paper, which they presented to Zaremba for publication.
     
   • The war delayed publication but the paper, Banach's first, appeared in the Bulletin of the Krakow Academy in 1918.
     
   • From the time that he produced these first results with Steinhaus, Banach started to produce important mathematics papers at a rapid rate.
     
   • Of course it is impossible to say whether, without the chance meeting with Steinhaus, Banach would have followed the route of research in mathematics.
     
   • It was also through Steinhaus that Banach met his future wife Lucja Braus.
     
   • Banach lectured to the Society twice during 1919 and continued to produce top quality research papers.
     
   • Banach was offered an assistantship to Lomnicki at Lvov Technical University in 1920.
     
   • This was, of course, not the standard route to a doctorate, for Banach had no university mathematics qualifications.
     
   • In 1922 the Jan Kazimierz University in Lvov awarded Banach his habilitation for a thesis on measure theory.
     
   • The University Calendar for 1921-22 reports [The life of Stefan Banach (Boston, 1996).',3)">3]:- .
     
   • On 7 April 1922, by resolution of the Faculty Council, Dr Stefan Banach received his habilitation for a Docent of Mathematics degree.
     
   • In 1924 Banach was promoted to full professor and he spent the academic year 1924-25 in Paris.
     
   • The years between the wars were extremely busy one for Banach.
     
   • In 1929, together with Steinhaus, he started a new journal Studia Mathematica and Banach and Steinhaus became the first editors.
     
   • These were set up under the editorship of Banach and Steinhaus from Lvov and Knaster, Kuratowski, Mazurkiewicz, and Sierpinski from Warsaw.
     
   • The first volume in the series Theorie des Operations lineaires was written by Banach and appeared in 1932.
     
   • In 1936 Banach gave a plenary address at the International Congress of Mathematicians in Oslo.
     
   • Another important influence on Banach was the fact that Kuratowski was appointed to the Lvov Technical University in 1927 and worked there until 1934.
     
   • Banach collaborated with Kuratowski and they wrote some joint papers during this period.
     
   • The way that Banach worked was unconventional.
     
   • It was difficult to outlast or outdrink Banach during these sessions.
     
   • The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed.
     
   • Andrzej Turowicz, also a professor of mathematics at the an Kazimierz University in Lvov, also described Banach's style of working (see [The life of Stefan Banach (Boston, 1996).',3)">3]):- .
     
   • [Banach] would spend most of his days in cafes, not only in the company of others but also by himself.
     
   • In 1939, just before the start of World War II, Banach was elected as President of the Polish Mathematical Society.
     
   • Banach had been on good terms with the Soviet mathematicians before the war started, visiting Moscow several times, and he was treated well by the new Soviet administration.
     
   • Banach's father came to Lvov fleeing from the German armies advancing towards Krakow.
     
   • Life at this stage was little changed for Banach who continued his research, his textbook writing, lecturing and sessions in the cafes.
     
   • Sobolev and Aleksandrov visited Banach in Lvov in 1940, while Banach attended conferences in the Soviet Union.
     
   • The Nazi occupation of Lvov in June 1941 meant that Banach lived under very difficult conditions.
     
   • Towards the end of 1941 Banach worked feeding lice in German institute dealing with infectious diseases.
     
   • As soon as the Soviet troops retook Lvov Banach renewed his contacts.
     
   • Sobolev, giving an address at a memorial conference for Banach, said of this meeting (see for example [The life of Stefan Banach (Boston, 1996).',3)">3]):- .
     
   • Despite heavy traces of the war years under German occupation, and despite the grave illness that was undercutting his strength, Banach's eyes were still lively.
     
   • He remained the same sociable, cheerful, and extraordinarily well-meaning and charming Stefan Banach whom I had seen in Lvov before the war.
     
   • Banach planned to go to Krakow after the war to take up the chair of mathematics at the Jagiellonian University but he died in Lvov in 1945 of lung cancer.
     
   • Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces.
     
   • In his dissertation, written in 1920, he defined axiomatically what today is called a Banach space.
     
   • The name 'Banach space' was coined by Frechet.
     
   • Banach algebras were also named after him.
     
   • A Banach space is a real or complex normed vector space that is complete as a metric space under the metric .
     
   • The completeness is important as this means that Cauchy sequences in Banach spaces converge.
     
   • A Banach algebra is a Banach space where the norm satisfies .
     
   • The importance of Banach's contribution is that he developed a systematic theory of functional analysis, where before there had only been isolated results which were later seen to fit into the new theory.
     
   • Banach proved a number of fundamental results on normed linear spaces, and many important theorems are today named after him.
     
   • There is the Hahn-Banach theorem on the extension of continuous linear functionals, the Banach-Steinhaus theorem on bounded families of mappings, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball.
     
   • The Banach-Tarski paradox appeared in a joint paper of the two mathematicians in 1926 in Fundamenta Mathematicae entitled Sur la decomposition des ensembles de points en partiens respectivement congruent.
     
   • The Banach-Tarski paradox was a major contribution to the work being done on axiomatic set theory around this period.
     
   • Banach's open mapping of 1929 also uses set-theoretic concepts, this time concepts introduced by Baire in his 1899 dissertation.
     
   • Honours awarded to Stefan Banach .
     
   • http://www-history.mcs.st-andrews.ac.uk/Biographies/Banach.html .
     

2. Mazur biography
   
   • In [The life of Stefan Banach (Boston, 1996).',1)">1] an incident from this time is recalled by Andrzej Turowicz.
     
   • He became a student of Banach's who taught at the university in Lvov.
     
   • His doctorate, under Banach's supervision, was awarded in 1935.
     
   • Mazur was a close collaborator with Banach at Lvov and became a member of the Lvov School of Mathematics, a group of about a dozen mathematicians working in functional analysis, real functions and probability theory.
     
   • He wrote several papers in collaboration with Banach during the 1930s and, having a better knowledge than Banach of German, he polished the language used in the joint papers which they wrote in German.
     
   • The collaboration between Mazur and Banach in Lvov was important for both men.
     
   • There is no doubt that of all Banach's colleagues in Lvov, Mazur was the one closest to him.
     
   • These sessions are described in [The life of Stefan Banach (Boston, 1996).',1)">1]:- .
     
   • Ulam, in [Adventures of a mathematician (New York, 1976).',2)">2], describes the particular way that the collaboration between Mazur and Banach in the Scottish Cafe worked:- .
     
   • The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed.
     
   • Mazur contributed 24 problems to the book with himself as the sole author, and a further 19 problems jointly contributed with others such as Banach.
     
   • It was not only Banach with who Mazur collaborated but others including Ulam.
     
   • Although it is not known for certain how the Scottish Book survived the war, we do know that it was brought to Wroclaw by Banach's wife and it ended up in the possession of his son.
     
   • The problem asked (although not in these words) about the existence of Schauder bases in separable Banach spaces.
     
   • Mazur made important contributions to geometrical methods in linear and non-linear functional analysis and to the study of Banach algebras.
     
   • The mean ergodic theorem in Banach spaces was announced by Mazur in 1932 but a proof does not appear in print until 1938 when Yosida and by Kakutani published the result.
     
   • 277 (3) (1987), 489-528.',4)">4] points out how many of Mazur's original contributions are not explicitly identified as such but appear in print only as remarks in Banach's Theorie des operations lineaires.
     
   • For example, the weak-basis theorem, due to Mazur, is given by Banach in his book but no proof appears.
     
   • The award was made in recognition that he was a leading Polish mathematician and a cofounder with Banach of the Polish School of Functional Analysis.
     

3. Luchins biography
   
   • At Oregon her thesis advisor was Bertram Yood, and Luchins submitted her thesis On Some Properties of Certain Banach Algebras in 1957.
     
   • In 1958 her fifth child was born and in the following year the two papers On radicals and continuity of homomorphisms into Banach algebras and On strictly semi-simple Banach algebras appeared, both in the Pacific Journal of Mathematics.
     
   • A Banach algebra is said to be absolute if every homomorphism of a Banach algebra into it is continuous, and is said to be strictly semi-simple if its two-sided regular maximal right ideals have zero intersection.
     
   • It is proved that an absolute Banach algebra contains no non-zero nilpotent elements, and that a strictly semi-simple Banach algebra is absolute.
     
   • For certain special Banach algebras (including semi-simple annihilator algebras) it is proved that if B contains no non-zero nilpotent elements, then B is strictly semi-simple (and hence absolute).
     
   • An example of a sss Banach algebra is the real algebra C(X,Q) of all quaternion-valued functions which are continuous and vanish at infinity on the locally compact Hausdorff space X.
     
   • It is proved that a real Banach algebra is sss if and only if it is isomorphic with a subalgebra of C(X,Q).
     
   • Thus any subalgebra (closed or not) of a sss real Banach algebra is sss.
     
   • Finally it is proved that the strict radical of a real Banach algebra contains the set of topologically nilpotent elements.
     
   • Under the natural norm (the sup norm) and under the spectral radius norm r(f), which is equivalent to the sup norm, C(X,Q) is a Banach algebra.
     
   • The authors announce some results concerning the closure of the set of nilpotent operators on a Banach space.
     

4. Orlicz biography
   
   • In 1919 Orlicz's family moved to Lvov (Lwow in Polish), where he completed his secondary education and then studied mathematics at the Jan Kazimierz University in Lvov having Stefan Banach, Hugo Steinhaus and Antoni Lomnicki as teachers.
     
   • Working in Lvov Orlicz participated in the famous meetings at the Scottish Cafe (Kawiarnia Szkocka) where Stefan Banach, Hugo Steinhaus, Stanislaw Ulam, Stanislaw Mazur, Marek Kac, Juliusz Schauder, Stefan Kaczmarz and many others talked about mathematical problems and looked for their solutions.
     
   • In Lvov under the leadership of our dear masters Banach and Steinhaus we were practising intricacies of mathematics.
     
   • When in 1960 Steinhaus was writing about Banach he emphasised this fact (Nauka Polska 8 (4) (1960), 157 or Wiadom.
     
   • Mazur and Orlicz are direct pupils of Banach; they represent the theory of operations today in Poland and their names on the cover of "Studia Mathematica" indicate direct continuation of Banach's scientific program.
     
   • Orlicz was awarded many high state decorations, prizes as well as medals of scientific institutions and societies, including the Stefan Banach Prize of the Polish Mathematical Society (1948), the Golden Cross of Merit (1954), the Commander's Cross of Polonia Restituta Order (1958), Honorary Membership of the Polish Mathematical Society (1973), the Alfred Jurzykowski Foundation Award (1973), Copernicus Medal of the Polish Academy of Sciences (1973), Order of Distinguished Teacher (1977), Waclaw Sierpinski Medal of the Warsaw University (1979), Medal of the Commission for National Education (1983) and the Individual State Prizes (second degree in 1952, first degree in 1966).
     
   • Orlicz's contribution is important in the following areas in mathematics: function spaces (mainly Orlicz spaces), orthogonal series, unconditional convergence in Banach spaces, summability, vector-valued functions, metric locally convex spaces, Saks spaces, real functions, measure theory and integration, polynomial operators and modular spaces.
     
   • Orlicz spaces Lφ = Lφ (Ω, Σ, μ) are Banach spaces consisting of all x ∈ L0(Ω, Σ, μ) such that ∫Ω φ(λ|x(t)|)dμ(t) < ∞ for some λ = λ(x) > 0 with the Orlicz norm: .
     
   • monographs on Orlicz spaces: M A Krasnoselskii and Ya B Rutickii, Convex Functions and Orlicz Spaces (Groningen 1961), J Lindenstrauss and L Tzafriri, Classical Banach Spaces I, II (Springer 1977, 1979), C Wu and T Wang, Orlicz Spaces and their Applications, (Harbin 1983 - Chinese), A C Zaanen, Riesz Spaces II, (North-Holland 1983), C Wu, T Wang, S Chen and Y Wang, Theory of Geometry of Orlicz Spaces (Harbin 1986 - Chinese), L Maligranda, Orlicz Spaces and Interpolation, (Campinas 1989), M M Rao and Z D Ren, Theory of Orlicz Spaces (Marcel Dekker 1991) and S Chen, Geometry of Orlicz Spaces (Dissertationes Math.
     
   • For example, the Orlicz-Pettis theorem says that in Banach spaces the classes of weakly subseries convergent and norm unconditionally convergent series coincide.
     
   • .functional analysis owes its magnificient development to Banach and his students, especially to Mazur, Orlicz and Schauder.
     
   • 23(1981), 222-231 and Achievements of Polish Mathematicians in the Domain of Functional Analysis in the Years 1919 - 1951, and biographies of S Banach, S Kaczmarz, A Lomnicki, S Mazur, J P Schauder).
     

5. Steinhaus biography
   
   • The youngsters were Stefan Banach and Otto Nikodym.
     
   • Also at this time Steinhaus started a collaboration with Banach and their first joint work was completed in 1916.
     
   • Banach was by this time on the staff at Lvov and the school rapidly grew in importance.
     
   • by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory.
     
   • Steinhaus published his second joint paper with Banach in 1927 Sur le principe de la condensation des singularites.
     
   • In 1929, together with Banach, he started a new journal Studia Mathematica and Steinhaus and Banach became the first editors.
     
   • The series was set up under the editorship of Steinhaus and Banach from Lvov and Knaster, Kuratowski, Mazurkiewicz, and Sierpinski from Warsaw.
     
   • It will not be a bad record to leave behind, to have had Banach as the first and Lebesgue as the last doctoral candidate.
     
   • He did important work on functional analysis, but he himself described his greatest discovery in this area as Stefan Banach.
     

6. Allan Graham biography
   
   • I took Graham's Part III course on Banach algebras in the year 1966 - 67.
     
   • He published A note on B*-algebras (1965), A spectral theory for locally convex algebras (1965), On a class of locally convex algebras (1967), On one-sided inverses in Banach algebras of holomorphic vector-valued functions (1967), and Holomorphic vector-valued functions on a domain of holomorphy (1967).
     
   • He continued to publish two papers a year until 1971, these papers making deep and significant contributions to Banach algebras.
     
   • He was invited to contribute a survey article to the Bulletin of the London Mathematical Society and this survey Some aspects of the theory of commutative Banach algebras and holomorphic functions of several complex variables was published in 1971:- .
     
   • In this survey article the author outlines those parts of holomorphic function theory (in particular, the holomorphic functional calculus) that have been applied in the study of commutative Banach algebras, specifically excluding topics that are peculiar to uniform algebras, where the applications have been most extensive.
     
   • In the five years 1972 to 1976 inclusive he published four papers, one being Several complex variables and Banach algebras in 1976:- .
     
   • This paper is a nicely-written brief introduction to that part of the general theory of commutative Banach algebras in which complex function theory plays a significant role.
     
   • The author assumes no previous knowledge of Banach algebra theory, but some acquaintance with the beginnings of functional analysis is assumed as is the general notion of a holomorphic function of several complex variables.
     

7. Saks biography
   
   • Steinhaus recalls a contribution Saks made around the same time (see for example [The life of Stefan Banach (Boston, 1996).',2)">2]):- .
     
   • In 1927, a collaboration [between Steinhaus and] Banach resulted in a paper "Sur le principe de la condensation des singularites" published in Fundamenta Mathematicae 9.
     
   • Banach and Saks collaborated on a joint paper Sur la convergence forte dans le champ Lp.
     
   • Published in 1930 [The life of Stefan Banach (Boston, 1996).',2)">2]:- .
     
   • This gave birth to a class of spaces that are still actively studied and that are now called spaces with the Banach-Saks property.
     
   • In 1937 an English translation of Theory of the integral was published with Banach's article The Lebesgue integral in abstract spaces as an appendix.
     
   • There he worked with Banach in the Soviet held town for two years, being appointed a professor at Lvov University which had been renamed the Ivan Franko University by the Russians.
     
   • At this time he taught in the Department with Banach at its head.
     

8. Bishop biography
   
   • Returning to Chicago in 1952 Bishop submitted his doctoral thesis Spectral Theory for Operations on Banach Spaces in 1954.
     
   • Here Bishop worked on uniform algebras (commutative Banach algebras with unit whose norms are the spectral norms) proving results such as antisymmetric decomposition of a uniform algebra, the Bishop-DeLeeuw theorem, and the proof of existence of Jensen measures.
     
   • (3) Banach spaces and operator theory.
     
   • An examples of a paper by Bishop on this topic is Spectral theory for operators on a Banach space (1957).
     
   • Examples of Bishop's papers in this area are Analyticity in certain Banach spaces (1962).
     

9. Bourgain biography
   
   • Bourgain's work touches on several central topics of mathematical analysis: the geometry of Banach spaces, convexity in high dimensions, harmonic analysis, ergodic theory, and finally, nonlinear partial differential equations from mathematical physics.
     
   • In his work on Banach spaces, Bourgain has studied problems examining how large a section of a finite dimensional Banach space can look like a Hilbert subspace.
     
   • In 1989 he proved some remarkable results, using analytic and probabilistic methods, which solved the L(p) problem which had been a longstanding one in Banach space theory and harmonic analysis.
     

10. Helly biography
    
    • During this period, he undertook research on functional analysis and proved the Hahn-Banach theorem in 1912, fifteen years before Hahn published essentially the same proof and 20 years before Banach gave his new setting.
      
    • One is the fact that he gives the Hahn-Banach theorem for the space C[a, b], while he is providing a simpler proof of a theorem which Riesz had published the previous year.
      
    • He also gives the uniform boundedness principle for linear functionals, the Banach-Steinhaus theorem.
      

11. Stone biography
    
    • One of the most famous books in that subject is Stefan Banach's Theorie des operations lineaires - it too was published in 1932 (in Warsaw, when Banach was 30).
      
    • Banach begins with a chapter on Lebesgue integration followed by a chapter on metric spaces (but I couldn't find a mention of either Hilbert or Stone in his book).
      
    • (The only reference to Banach concerns a 1924 paper in the Fundamenta.) The work consists of ten chapters, of which the last (Applications) is more than a third of the book.
      

12. Johnson Barry biography
    
    • Johnson is well known for his work on Banach algebras and operator algebras, in particular, studying cohomology in these algebras.
      
    • His mathematical publications started in 1964 with a series of papers on topological algebras, measure algebras and Banach algebras.
      
    • In 1972 Johnson wrote a joint paper with Ringrose and Kadison on cohomology of operator algebras and in the same year his book Cohomology in Banach algebras (1972) appeared.
      

13. Hahn biography
    
    • However to many mathematicians he is best remembered for the Hahn-Banach theorem which we mention again below.
      
    • These include a report on integral equation he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927.
      

14. Dieudonne biography
    
    • For example, the differential calculus is developed in terms of linear approximation to functions on an open subset of a Banach space to a Banach space.
      
    • Yet it would be completely false to assert that the book contained a study of Banach spaces - no non-trivial proposition on such spaces is proved.
      

15. Zaanen biography
    
    • As a student, he came into contact with the ideas of modern analysis via Zygmund's book on trigonometric series and Banach's book on linear transformations.
      
    • Measure and integral, Banach and Hilbert space, linear integral equations (1953) which contained much of his own research as well as material from a lecture course by N G de Bruijn.
      
    • Measure and integral, Banach and Hilbert space, linear integral equations (1953), we have already mentioned above.
      

16. Riesz biography
    
    • In 1918 his work came close to an axiomatic theory for Banach spaces, which were set up axiomatically two years later by Banach in his dissertation.
      
    • Many of Riesz's fundamental findings in functional analysis were incorporated with those of Banach.
      

17. Ulam biography
    
    • from the Polytechnic Institute in Lvov in 1933 where he studied under Banach.
      
    • Banach in 1929 had solved a related measure problem, but assuming the Generalised Continuum Hypothesis.
      
    • Ulam, in 1930, strengthened Banach's result by proving it without using the Generalised Continuum Hypothesis.
      

18. Zaremba biography
    
    • He described Zaremba's teaching style (see [The life of Stefan Banach (Boston, 1996).',1)">1]):- .
      
    • Lebesgue, someone who seldom heaped praise on his colleagues, paid tribute to him in 1930 when Zaremba received an honorary degree from the Jagiellonian University in Krakow (see for example [The life of Stefan Banach (Boston, 1996).',1)">1] or [Half a century of Polish mathematics (Warsaw, 1973).',2)">2]):- .
      

19. Krein biography
    
    • During this time he worked on topics such as Banach spaces, the moment problem, integral equations and matrices, and on spectral theory for linear operators.
      
    • His work is the culmination of the noble line of research begun by Chebyshev, Stieltjes, Sergei Bernstein and Markov and continued by F Riesz, Banach and Szego.
      

20. Rennie biography
    
    • In any Banach lattice Lp , p > 1, the L-topology is the metric topology; in any Banach lattice B, it is the star topology.
      

21. Tarski biography
    
    • He published a joint paper with Banach in that year on what is now called the Banach-Tarski paradox.
      

22. Petryshyn biography
    
    • This outstanding reference/text develops an essentially constructive theory of solvability on linear and nonlinear abstract and differential equations involving A-proper operator equations in separable Banach spaces, treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications.
      
    • Facilitating the understanding of the solvability of equations in infinite dimensional Banach space through finite dimensional approximations, Approximation - solvability of Nonlinear Functional and Differential Equations: offers an important elementary introduction to the general theory of A-proper and pseudo-A-proper maps; develops the linear theory of A-proper maps; furnishes the best possible results for linear equations; establishes the existence of fixed points and eigenvalues for P-gamma-compact maps, including classical results; provides surjectivity theorems for pseudo-A-proper and weakly-A-proper mappings that unify and extend earlier results on monotone and accretive mappings; shows how Friedrichs' linear extension theory can be generalized to the extensions of densely defined nonlinear operators in a Hilbert space; presents the generalized topological degree theory for A-proper mappings; and applies abstract results to boundary value problems and to bifurcation and asymptotic bifurcation problems.
      

23. Kuratowski biography
    
    • At Lvov, however, Kuratowski worked with Banach and they answered some fundamental problems on measure theory.
      
    • Ulam, who had become Banach's research student also worked with them.
      

24. Brown Gavin biography
    
    • Further papers, all arising from his doctoral studies, followed in quick succession: Stability of wedges and semi-algebras (1968); Type 0 semi-algebras in Banach algebras (1968); Continuous functions of bounded n th variation (1969); and Norm properties of a class of semi-algebras (1969).
      
    • A semi-algebra in a Banach algebra B is a subset A of B such that x + y, αx and x.y belong to A whenever x and y are in A, and α ≥ 0 is real number.
      

25. Lopatynsky biography
    
    • Recently he has obtained important results on solvability of the Cauchy problem for operator equations in Banach space and also on "almost everywhere" solvability of general linear and nonlinear boundary problems.
      
    • He also obtained some basic results in the solvability of the Cauchy problem for operator equations in Banach spaces.
      

26. Arino biography
    
    • Stabilite d'un ensemble ferme pour une equation differentielle a argument retarde (1978); "Our aim is to establish a local existence result for a differential equation with delay in a reflexive Banach space, with the hypothesis of weak continuity in the second member.
      
    • Comportement des solutions d'equations differentielles a retard dans un espace ordonne (1980); "Using vectorial Ljapunov functionals, we give here some results related to the behaviour at infinity of solutions of a differential equation with delay in an ordered Banach space." .
      

27. Schauder biography
    
    • Schauder published fixed point theorems for Banach spaces in 1930.
      
    • Schauder's main achievement consists in transferring some topological notions and theorems to Banach spaces (the fixed point theorem, invariance of domain, the concept of index).
      

28. Karlin biography
    
    • He was appointed to the California Institute of Technology in 1948 and began publishing papers on functional analysis such as Unconditional convergence in Banach spaces (1948), Bases in Banach spaces (1948), Orthogonal properties of independent functions (1949), and (with L S Shapley) Geometry of reduced moment spaces (1949).
      

29. Kakutani biography
    
    • He read various classic texts including those of Stone and Banach and by the time of his graduation at the end of the three year course he had a good foundation in modern analysis.
      
    • Among the areas on which he has written papers we must mention: complex analysis, topological groups, fixed point theorems, Banach spaces and Hilbert spaces, Markov processes, measure theory, flows, Brownian motion, and ergodic theory.
      

30. Smithies biography
    
    • He was influenced, by reading books by Banach and Stone, and attending lectures by Courant and von Neumann, to become interested in functional analysis despite Hardy's dislike of abstract mathematics.
      
    • Smithies looks in detail at the development of the concept of an adjoint operator in the years before the Hahn-Banach theorem.
      

31. Northcott biography
    
    • At this point Northcott was awarded a Commonwealth Fund Scholarship to allow him to study Banach spaces at Princeton University.
      
    • Sometimes he tried to reconstruct proofs of results that he had learnt as a student; at other; he attempted to build up a theory of integration for functions with values in a Banach space.
      

32. Caccioppoli biography
    
    • In the same year Caccioppoli considered the extension of the definition of linear functionals from the set of continuous functions to the set of Baire functions, anticipating a special case of the Hahn-Banach theorem.
      
    • To decide on both existence and uniqueness (and not only on existence, as Brouwer's theorem does) he provided the general concept of functional correspondence inversion, stating, in 1932, that a transformation between two Banach spaces is invertible only if it is locally invertible and if the compact sequences are the only ones to be transformed into convergent sequences.
      

33. Zorn biography
    
    • In 1945 he published the paper Characterization of analytic functions in Banach space in the Annals of Mathematics.
      
    • The concept of analyticity may be extended in various ways to functions from one complex Banach space to another.
      

34. Krasnosel'skii biography
    
    • For example Positive solutions of operator equations (1962) which studied the existence, uniqueness, and properties of positive solutions of linear and non-linear equations in a partially ordered Banach space, Vector fields in the plane (1963) which the angular variation of a plane vector field relative to a curve, and Displacement operators along trajectories of differential equations (1966) which is described by C Olech as follows:- .
      
    • is a remarkable and important book, treating the theory and application of ordered Banach spaces and positive (linear and nonlinear) operators.
      

35. Copson biography
    
    • to problems in classical algebra and analysis show how much can be done without ever defining a normed vector space, a Banach space or a Hilbert space.
      

36. Toeplitz biography
    
    • In the 1930s he developed a general theory of infinite dimensional spaces and criticised Banach's work as being too abstract.
      

37. Goldstine biography
    
    • I (1940), written jointly with R P McKeon, and Linear functionals and integrals in abstract spaces (1942) in which he shows that the Daniell integral and the integral similarly defined by Banach in his addendum to Saks's "Theory of the Integral" are Lebesgue integrals with respect to regular Caratheodory outer measures.
      

38. Ostrowski biography
    
    • By 1973 the third edition of this monograph appeared, this time with a new title: Solution of equations in Euclidean and Banach spaces.
      

39. Aleksandrov biography
    
    • Many famous foreign mathematicians also visited Komarovka - Hadamard, Frechet, Banach, Hopf, Kuratowski, and others.
      

40. Janiszewski biography
    
    • He wrote in his article (see for example [The life of Stefan Banach (Boston, 1996).',2)">2]):- .
      

41. Naimark biography
    
    • Naimark also contributed to Banach spaces.
      

42. Feigenbaum biography
    
    • He again taught, giving courses on Banach spaces and C*-algebras.
      

43. Gelfand biography
    
    • Gelfand's theory of normed rings revealed close connections between Banach's general functional analysis and classical analysis.
      

44. Peres biography
    
    • By the time the work was published the ideas it contained were no longer in the mainstream of development of functional analysis since topological and algebraic concepts introduced by Banach, von Neumann, Stone and others were determining the direction of the subject.
      

45. Plessner biography
    
    • It seems that his interest in functional analysis arose when he read Banach's book of 1932.
      

46. Keen biography
    
    • In the early 1960s, Bers and Ahlfors showed that the space of conformal structures on a given Riemann surface can be modelled on a Banach space with a real analytic structure.
      

47. Silva biography
    
    • This paper appears to be inspired in part by the work of Pincherle and Fantappie, but it is written more in the recent style of functional analysis in which abstract spaces, particularly Banach spaces, play a central role.
      

48. Iwanik biography
    
    • He submitted a dissertation Extreme operators on classical Banach spaces to the Scientific Council of the Institute of Mathematics of the Polish Academy of Sciences and was awarded an habilitation degree in 1978.
      

49. Leray biography
    
    • After his 1934 paper with Schauder, Leray published a paper on algebraic topology in the following year on the topology of Banach spaces.
      

50. Zorawski biography
    
    • Among the sixteen mathematicians present were Stefan Banach, Otto Nikodym, Stanislaw Zaremba, and Kazimierz Zorawski.
      

51. Livsic biography
    
    • His habilitation thesis on generalisations of von Neumann's extension theory was examined in 1945 by a powerful groups of mathematicians, namely Banach, Gelfand, Naimark and Plessner at the Steklov Institute.
      

52. Schatten biography
    
    • Schatten's principal mathematical achievement was that of initiating the study of tensor products of Banach spaces.
      

53. Gnedenko biography
    
    • There he met Banach and [J.
      

54. Tapia biography
    
    • In it Tapia considered the solution of the equation P(x) = 0, where P is a nonlinear mapping between Banach spaces.
      

55. Kato biography
    
    • Transformations in Banach spaces are also considered when they can be treated as generalizations of self-adjoint transformations.
      

56. Rado biography
    
    • However unlike Rado, who had only just begun his university studies, Helly was already a research mathematician who had made remarkable progress in his work on functional analysis, proving the Hahn-Banach theorem in 1912.
      

57. Schwartz biography
    
    • This has led to extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables.
      

58. Herstein biography
    
    • The second paper proves a conjecture that the solubility of groups of odd order is equivalent to a condition on the group ring of a group, while the third paper takes methods from the study of Banach rings and topological groups to prove results about group rings over the complex numbers.
      

59. Kolmogorov biography
    
    • Many famous mathematicians visited Komarovka: Hadamard, Frechet, Banach, Hopf, Kuratowski, and others.
      

60. Kaplansky biography
    
    • He completed the solution of Kurosh's problem on algebraic algebras of bounded degree, where Jacobson had made a decisive reduction, and considered numerous questions in the area of Banach algebras, always from the algebraist's viewpoint.
      

61. Wolf Frantisek biography
    
    • Frank's most influential papers were probably those concerned with the analytical perturbation of operators on Banach spaces and his studies on the essential spectrum of certain singular elliptic differential operators.
      

62. Petersson biography
    
    • the abstract notion of a vector space appeared in print only in 1922 in papers by S Banach and H Hahn.
      

63. Eilenberg biography
    
    • It was there that Eilenberg met Banach, who led the Lvov mathematicians.
      

64. Kober biography
    
    • Kober was a highly productive mathematician working on special functions, functional analysis (in this area Kober's Theorem which appeared A theorem on Banach spaces (1939) is named after him), approximation theory and the theory of functions of a real variable.
      

65. Straus biography
    
    • An earlier joint paper by Straus and Erdos was On linear independence of sequences in a Banach space (1953).
      

66. Fenyo biography
    
    • The matters discussed are metric and normed spaces with particular reference to Hilbert spaces, Hahn-Banach theory, operators (including inverse, dual and compact operators) and eigenvalues and eigenvectors.
      

67. Nikodym biography
    
    • the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory and the foundations of quantum mechanics.
      

68. Hirzebruch biography
    
    • He has been awarded many prizes: the Silver Medal from the Swiss Federal Institute of Technology (1950), the Wolf Prize for Mathematics (1988), the Lobachevsky Prize from the Russian Academy of Sciences (1989), the Seki-Takakazu Prize of the Japanese Mathematical Society (1996), the Cothenius Gold Medal Leopoldina 1997, the Lomonosov Gold Medal of the Russian Academy of Sciences (1997), the Albert Einstein Medal (1999), the Stefan Banach Medal of the Polish Academy of Sciences (2000), the Krupp-Wissenschaftspreiss (2000), the Helmholtz Medal of the Berlin-Brandenburg Academy of Sciences 2002, the Georg Cantor Medal of the German Mathematical Society (2004).
      

69. Rogers biography
    
    • His later work covered a wide range of different topics in geomery and analysis including Borel functions, Hausdorff measure and local measure, topological properties of Banach spaces and upper semicontinuous functions.
      

History Topics

1. The Scottish Book
   
   • If I remember correctly, it was S Banach who suggested keeping track of some of the problems occupying the group of mathematicians there The mathematical life was very intense in Lwow Some of us met practically every day, informally in small groups, at all times of the day to discuss problems of common interest, communicating to each other the latest work and results.
     
   • A large notebook was purchased by Banach and deposited with the headwaiter of the Scottish Coffee House, who, upon demand, would bring it out of some secure hiding place, leave it at the table, and after the guests departed, return it to its secret location.
     
   • According to Steinhaus, this document was brought back to the city of Wroclaw by Banach's son, now a physician in Poland.
     

2. Topology history
   
   • A further step in abstraction was taken by Banach in 1932 when he moved from inner product spaces to normed spaces.
     Go directly to this paragraph
     
   • Banach took Frechet's linear functionals and showed that they had a natural setting in normed spaces.
     Go directly to this paragraph
     

3. Abstract linear spaces
   
   • Like so much work in this area it had very little immediate impact and axiomatic infinite dimensional vector spaces were not studied again until Banach and his associates took up the topic in the 1920's.
     Go directly to this paragraph
     
   • The fully axiomatic approach appeared in Banach's 1920 doctoral dissertation.
     Go directly to this paragraph
     

Famous Curves

Societies etc

1. European Mathematical Society Prizes
   
   • whose work has made the geometry of Banach spaces look completely different.
     
   • To mention some of his spectacular results: he solved the notorious Banach hyperplane problem, to find a Banach space which is not isomorphic to any of its hyperplanes.
     
   • He gave a counterexample to the Schroder-Bernstein theorem for Banach spaces.
     
   • He proved a deep dichotomy principle for Banach spaces which if combined with a result of Komorowski and Tomczak-Jaegermann shows that if all closed infinite-dimensional subspaces of a Banach space are isomorphic to the space, then it is a Hilbert space.
     
   • He gave (jointly with Maurey) an example of a Banach space such that every bounded operator from the space to itself is a Fredholm operator.
     
   • He solved a problem by Johnson and Lindenstrauss on embeddings of finite metric spaces into Banach spaces.
     
   • He also obtained sharp results on almost isometric embeddings of finite dimensional Banach spaces using uniform distributions of points on spheres.
     
   • The proof involves several remarkable technical and conceptual developments, like a bivariant K-theory for Banach algebras (versus Kasparov's by now classical one for C*-algebras) or establishing the conjecture for various completions of the L1 algebras of the groups.
     

2. Polish Mathematical Society
   
   • The youngsters were Stefan Banach and Otto Nikodym.
     
   • Among the sixteen mathematicians present were Stefan Banach, Otto Nikodym, Stanislaw Zaremba, and Kazimierz Zorawski.
     
   • In 1939, just before the start of World War II, Banach was elected as President of the Society.
     

3. BMC 1973
   
   • Lance, CInjective Banach spaces and C*-algebras .
     

4. Young Mathematician prize
   
   • for works on the theory of stochastic differential equations and Banach geometry.
     
   • for an approximation p-th order property of a Banach space.
     

5. BMC 1976
   

6. BMC 1969
   
   • Duncan, JRelations between algebras and geometry in Banach algebras .
     

7. BMC 1977
   
   • Johnson, B EStability of Banach algebras .
     

8. BMC 1981
   
   • Gillespie, T ASelf-adjoint operators on Banach spaces .
     

9. BMC 1971
   
   • Allan, G RAnalytic functions and Banach algebras .
     

10. BMC 1979
    
    • Craw, I GCommutative Banach algebras and the topology of the maximal ideal space .
      

11. BMC 2004
    
    • Dineen, S Banach-valued spectra .
      

12. BMC 1993
    
    • Haydon, R G Smooth functions and equivalent norms on Banach space .
      

13. BMC 1972
    

14. BMC 1975
    
    • Taylor, J LSome invariants associated with Banach algebras .
      

15. BMC 1980
    
    • Dales, H GRadical Banach algebras .
      

16. Polish Academy of Sciences
    
    • In 1972 the International Stefan Banach Centre was established as part of the Institute of Mathematics of the Polish Academy of Sciences.
      

17. International Congress Speakers
    
    • Stefan Banach, Die Theorie der Operationen und ihre Bedeutung fur die Analysis.
      

18. AMS Steele Prize
    
    • for two seminal papers "Viscosity solutions of Hamilton-Jacobi equations" (joint with P-L Lions), and "Generation of semi-groups of nonlinear transformations on general Banach spaces" (joint with T M Liggett).
      

19. BMC 1982
    
    • Mulvey, CGlobalising the Hahn-Banach theorem .
      

20. BMC 1968
    
    • Johnson, B ERelations between algebras and topology in Banach algebras .
      

21. BMC 1995
    
    • Gowers, W T Ramsey theory, games and the structure of Banach space .
      

22. BMC 1978
    
    • Phelps, R R Banach spaces with the Radon-Nikodym property; convex sets and convex functions: some remarkable relationships .
      

References

1. References for Banach
   
   • References for Stefan Banach .
     
   • R Kaluza, The life of Stefan Banach (Boston, 1996).
     
   • F Baranski, Lvovian reminiscences of Stefan Banach (Polish), Opuscula Math.
     
   • K Ciesielski, Some facts about the life of Stefan Banach (Polish), Opuscula Math.
     
   • K Ciesielski, On some details of Stefan Banach's life, Opuscula Math.
     
   • Geburtstag von Stefan Banach, Wiss.
     
   • G Hoare and N Lord, Stefan Banach (1892-1945): A commemoration of his life and work, The Mathematical Gazette 79 (1995), 456-470.
     
   • S Lal, Cardan and Banach : a comparative study, Math.
     
   • E Marczewski, Sur l'oeuvre scientifique de Stefan Banach II.
     
   • S M Nikolski, Reminiscences of Stefan Banach (Polish), Wiadomosci matematyczne 30 (1) (1993), 115-120.
     
   • Obituary: Stefan Banach (Russian), Uspekhi Matem.
     
   • Obituary: Stefan Banach, Colloquium Math.
     
   • W Orlicz, Sur l'oeuvre scientifique de Stefan Banach I.
     
   • H Steinhaus, Stefan Banach, Studia Mathematica 1 (1963), 7-15.
     
   • H Steinhaus, Souvenir de Stefan Banach, Colloquium Math.
     
   • H Steinhaus, Stefan Banach, 1892-1945, Rev.
     
   • H Steinhaus, Stefan Banach, Studia Math.
     
   • K Szalajko, Reminiscences of Stefan Banach against the background of Lvov and the Lvovian school of mathematics (Polish), Opuscula Math.
     
   • W Zelazko, Stefan Banach (1892-1945), European Mathematical Society Newsletter 5 (1992), 23.
     
   • http://www-history.mcs.st-andrews.ac.uk/References/Banach.html .
     

2. References for Zaanen
   
   • Curriculum vitae of A C Zaanen, in Operator theory in function spaces and Banach lattices (Birkhauser, Basel, 1995), 12-14.
     
   • List of publications of A C Zaanen, in Operator theory in function spaces and Banach lattices (Birkhauser, Basel, 1995), 7-11.
     

3. References for Tarski
   
   • A Kirsch, Das Paradoxon von Hausdorff, Banach und Tarski : Kann man es 'verstehen'?, Math.
     
   • J Pla Carrera, The axiom of choice and the Banach-Tarski paradox (Catalan), Butl.
     

4. References for Mazur
   
   • R Kaluza, The life of Stefan Banach (Boston, 1996).
     

5. References for Saks
   
   • R Kaluza, The life of Stefan Banach (Boston, 1996).
     

6. References for Ulam
   
   • J M Rassias, Stefan Banach, Alexander Markowic Ostrowski, Stanislaw Marcin Ulam, in Functional analysis, approximation theory and numerical analysis (River Edge, NJ, 1994), 1-4.
     

7. References for Janiszewski
   
   • R Kaluza, The life of Stefan Banach (Boston, 1996).
     

8. References for Zaremba
   
   • R Kaluza, The life of Stefan Banach (Boston, 1996).
     

9. References for Ricci Giovanni
   
   • M Cugiani, Commemoration of Giovanni Ricci (Italian), Geometry of Banach spaces and related topics, Milan, 1983, Rend.
     

10. References for Cardan
    
    • S Lal, Cardan and Banach : a comparative study, Math.
      

11. References for Helly
    
    • H Hochstadt, Eduard Helly, Father of the Hahn-Banach Theorem, The Mathematical Intelligencer 2 (1980), 123-125.
      

12. References for Steinhaus
    
    • R Kaluza, The life of Stefan Banach (Boston, 1996).
      

13. References for Ostrowski
    
    • J M Rassias, Stefan Banach, Alexander Markowic Ostrowski, Stanislaw Marcin Ulam, in Functional analysis, approximation theory and numerical analysis (River Edge, NJ, 1994), 1-4.
      

Additional material

1. Jacobson: 'Structure of Rings
   
   • The first, class includes the rings of bounded operators in Banach spaces.
     

Quotations

1. Quotations by Banach
   
   • Quotations by Stefan Banach .
     
   • http://www-history.mcs.st-andrews.ac.uk/Quotations/Banach.html .
     

Chronology

1. Mathematical Chronology
   
   • Banach is awarded his habilitation for a thesis on measure theory.
     
   • Banach and Tarski publish the "Banach-Tarski paradox" in a joint paper in Fundamenta Mathematicae: Sur la decomposition des ensembles de points en parties respectivement congruentes.
     
   • Bourgain, using analytic and probabilistic methods, solves the L(p) problem which had been a longstanding one in "Banach space" theory and harmonic analysis.
     

2. Chronology for 1920 to 1930
   
   • Banach is awarded his habilitation for a thesis on measure theory.
     
   • Banach and Tarski publish the "Banach-Tarski paradox" in a joint paper in Fundamenta Mathematicae: Sur la decomposition des ensembles de points en parties respectivement congruentes.
     

3. Chronology for 1980 to 1990
   
   • Bourgain, using analytic and probabilistic methods, solves the L(p) problem which had been a longstanding one in "Banach space" theory and harmonic analysis.