Mathematics Newsletter Volume 21. No4, March 2012
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MATHEMATICS NEWSLETTER EDITORIAL BOARD S. Ponnusamy (Chief Editor) Department of Mathematics Indian Institute of Technology Madras Chennai - 600 036, Tamilnadu, India Phone : +91-44-2257 4615 (office) +91-44-2257 6615, 2257 0298 (home) [email protected] http://mat.iitm.ac.in/home/samy/public_html/index.html S. D. Adhikari G. K. Srinivasan Harish-Chandra Research Institute Department of Mathematics, (Former Mehta Research Institute ) Indian Institute of Technology Chhatnag Road, Jhusi Bombay Allahabad 211 019, India Powai, Mumbai 400076, India [email protected] [email protected] C. S. Aravinda B. Sury, TIFR Centre for Applicable Mathematics Stat-Math Unit, Sharadanagar, Indian Statistical Institute, Chikkabommasandra 8th Mile Mysore Road, Post Bag No. 6503 Bangalore 560059, India. Bangalore - 560 065 [email protected], [email protected] [email protected] M. Krishna G. P. Youvaraj The Institute of Mathematical Sciences Ramanujan Institute CIT Campus, Taramani for Advanced Study in Mathematics Chennai-600 113, India University of Madras, Chepauk, [email protected] Chennai-600 005, India [email protected] Stefan Banach (1892–1945) R. Anantharaman SUNY/College, Old Westbury, NY 11568 E-mail: rajan−[email protected] To the memory of Jong P. Lee Abstract. Stefan Banach ranks quite high among the founders and developers of Functional Analysis. We give a brief summary of his life, work and methods. Introduction (equivalent of middle/high school) there. Even as a student Stefan revealed his talent in mathematics. He passed the high Stefan Banach and his school in Poland were (among) the school in 1910 but not with high honors [M]. He went then to major initiators and developer of Banach Space Theory, the city of Lwow and was enrolled in the Engineering dept; especially Geometric Functional Analysis. We give a short he graduated in 1914. (From hearsay) he supported himself by description of this remarkable mathematician of the first half of private tutoring. He returned to Krakow and earned money by the 20th century. He is a sort of “mathematical hero” to many, teaching in High Schools. among whom this author counts himself a humble member. We cover only a few aspects of Banach’s Life as in the list Discovered by Steinhaus below: We come now to a more interesting part and chance encounter §1. Youth. His talent discovered by Hugo Steinhaus. that changed the life of our hero. Hugo Steinhaus had been (in §2. Thesis and the Theorie´ des Operations Lineaires. The three 1916) appointed [M] to the university in the town of Lwow; main Theorems of Banach. however, he was then in Krakow. The professor was taking §3. Professor of mathematics at Lwow; teaching style. his regular walk one evening in the park when he saw ([G], §4. Working with colleagues and students; Scottish Problem [M]) two (shabbily dressed) young men seated on a bench. book-the intensity of research. They were discussing the Lebesgue Integral (!) which was §5. Word war II life under the Nazis; Death; Basis Problem. rather new. As the good professor was surprised greatly he §6. Conclusion, Hugo Steinhaus’ Banach Matchstick talked to them [M]. One of them was Otto Nikodym and problem. the other Stefan Banach who (from hearsay) was working in a steel mill. Steinhaus asked them to come to the Uni- The main sources for this paper are those by Ulam [U1], versity at Lwow. Banach worked on problems, solved them [U2] and [K-K]; we have also used the MacTutor [M] at the and published papers, including one authored jointly with university of St. Andrews, Scotland. The author is grateful to Steinhaus [M]. He submitted a thesis to the University; it made Prof. Som Naimpally for supplying the web link [M]; this has an exception to the rules so as to allow him to submit a thesis more details (especially Banach’s childhood) than what we (1922) with “no proper university training” [M]. As an aside give here. another self-taught mathematician was also lucky. Ramanujan §1. Youth was discovered by Hardy in England; E. Galois had no luck when alive. It is only fair to say that Stefan Banach’s childhood and youth were not ordinary. His mother left soon after he was bap- §2. Thesis and Theorie´ des Operations Lineaires tized [M]. He was brought up first by his grandmother, then by a kind lady in the city of Krakow; he learnt French there Stefan Banach’s thesis at the Univ. of Lwow was in (what we [M]. He attended primary school and then the Gymnasium now call) Banach Spaces. He himself called them “B-spaces” Mathematics Newsletter -113- Vol. 21 #4, March 2012 & Vol. 22 #1, June 2012 in [B]. Let us recall that a vector space E over the reals with a §4. Working with Colleagues & Students. (compation) norm ·is called a normed (vector) space. It is Scottish Book ([U], [K-K]) called a Banach space if cauchy sequences converge in this norm (to an element of E), i.e. (E, this must be the notation of Stefan Banach’s method to do research was to join colleagues norm) is a complete metric space. But a Banach space is much and students in evenings at local cafes:´ (Scottish Cafe´ & Rome more than just a complete metric space. Cafe).´ The “three basic principles of Linear Analysis” are ([Du-S]): At first the participants used the table top to write (math). Later at night the waiter would erase these writings. But (i) Banach-Steinhaus Theorem (also called Uniform bound- Banach’s wife, Lucille put an end to this wastefulness [K-K]. edness principle) She gave the waiter a cloth bound note book. Any mathe- (ii) Open Mapping Theorem & the Closed Graph Theorem matician could write problems in it. Usually a prize for solu- (iii) Hahn-Banach extension theorem. tion was also record like a bottle of wine, or a goose [K-K]. Completeness is crucial to (i) and (ii) but not to (iii) [Ru1]. The waiter would ceremoniously bring the note book out (for The result (i) is in Banach’s Thesis or soon thereafter. The problem-writing). This note book came to be known as the result (iii) was independently discovered by Hans Hahn in Scottish Problem book ([U], [K-K]). Germany. All of these were proved in the 1930’s or even Going out of historical order let us record a curious fact before. [U] about this note book. In the 1930 Ulam was staying The monograph “Theorie des Operations Lineaires” [B] is in USA and (naturally) visited Poland some times. On one dedicated to Banach’s wife Lucille Banach. This work is the such visit he and his friend and colleague there, S. Mazur first in Geometric Functional Analysis. In it, Banach develops thought there may be danger to the note book. They decided the general theory of Banach Spaces, and examines the under- that if the need arose, the Scottish Book was to be buried lying geometric structure of these classical spaces (among at a certain corner known only to them, in the local football others): field [U]. I think this ([0, 1]) should be along with C0. Of course not Returning to Banach and his school: he evidently enjoyed all results in [B] are due to Stefan Banach but “his contributions talking to friends and students – not just math. They talked are many and deep” ([Du-S], ch 2). Already in 1905 F. Riesz also of the changing face of Germany [U] and the emergence had identified the dual space of C[0, 1] as the space of all finite, of Nazis. It seems to this author that they did not forsee the regular measure on Borel subsets of [0, 1]. seriousness of the situation in future for Poland. §3. Prof. of Math at Lwow: Teaching Intensity of research Stefan Banach was Professor of math at the University of Again, returning to the meetings at the Cafe:´ According to Lwow from the early 1920’s. According to Ulam [U], Banach’s Ulam [U] these sittings sometimes went late into the night. lectures were not always well prepared, but it was a pleasure The next morning Banach would appear with slips of paper to see him get out of a difficulty (in math.) Apparently, Banach containing solutions, or ideas on how to solve them. If the was good in “thinking on his feet”. proofs were not quite complete or even quite correct, Banach’s As for his writing style it was fluent and free-flowing (as (“pet”) student Stanislaw Mazur would fix them [U]. The in [B]) i.e., not excessively formal, as in modern works. For other student was W. Orlicz. Ulam [U] recalls one ses- a masterly presentation by Banach, we refer to an appendix sion at the cafe´ that lasted non-stop for 17 hours (!) with in the classic gem [S] by S. Saks; we find Haar’s measure for breaks only for bathing and eating. From hearsay, it seems compact metric groups exposed there. that the result is the following classic, the Banach–Mazur With H. Steinhaus, Banach started the journal, Studia Theorem: Mathematica devoted to works in Functional Analysis. It is Every separable Banach space is isometric to a (closed) alive and kicking today. subspace of C[0, 1]; it is also a quotient of the space l1. Mathematics Newsletter -114- Vol. 21 #4, March 2012 & Vol. 22 #1, June 2012 Banach’s genius was much more creative than just Schauder basis problem ([B], 1932); its solution ([E], 1972): logical. Very few proofs of his results were incorrect, and Let us recall from Linear Algebra that a (finite) set of vectors were fixed by others later.