Steinhaus Dla HSC

<Steinhaus dla HSC>

Roman Duda

Steinhaus' formative years

Hugo Steinhaus was born in a well-to-do family in 1887 inJasło, a small town in Carpathian Mountains[1]. The region is picturesque and Steinhaus liked it very much. In his later yearshe has been willingly returning there or a rest. In 1905 he obtained in Jasło,in the local classic gymnasium, his matura exam "with a distinction". Mathematically talented, he went then to Lvov, a university city nearby, to study mathematics and philosophy under Professors Józef Puzyna and Kazimierz Twardowski. After a year, however, a friend of the Steinhaus family, Stanisław Jolles, himself a professor at the Berlin polytechnic (Charlottenburg), has persuaded the young student to move to Göttingen, an excellent mathematical centre in those days. Steinhaus followed the advice but he made in Göttingen a somewhat different choice. He decided to study pure mathematics and (instead of philosophy) applied mathematics, the latter under Professor Carl Runge. The route from Jasło to Göttingen led through Breslau/Wrocław and since Steinhaus has been returning home twice a year, for winter and summer holidays, he remembered the city well, never suspecting that he will spend here nearly three last decades of his life.

In Göttingen Steinhaus stayed five years 1906-1911. He attended lectures by, among others, Hilbert, Minkowski, Klein, Runge, Zermelo, Landau - the names speak for themselves. As it was customary for better students in those days, Steinhaus finished his studies by passing a rigorosum (on 10 May 1911) and presenting a thesis.

The thesis was entitled "New Applications of the Dirichlet Principle"[2]. The Dirichlet Principle consisted in solving a partial differential equation by its reduction to a problem in the calculus of variations. It was a way to prove general theorems and to obtain particular solutions as limits of a certain functional tending to its minimum. In particular, in the case of an equation

(1) 2u / x2 +2u / y2 = 0 in the region ,

it offered a way to find a differentiable solution u in which assumes given values at the boundary of . In the middle of XIX century it led to the following Minimum Problem: among all differentiable functions u(x,y) in , with the given boundary values, find that one for which the integral

(2)  [(u /x)2 + (u /y)2] dxdy



is minimal. Such a function solves the equation (1).

In that time all mathematicians, including Hilbert, were convinced that the Minimum Problem always has a solution and so that the problem (1) is essentially solved.

Steinhaus begun his thesis by showing, with a suitable counterexample, that this conviction is false. And then he proposed a new meaning of the Dirichlet Principleby extending it to any method which follows Bolzano-Weierstrass pattern of finding a concentration point and can be applied not only to numbers but also to functions and other mathematical objects and which leads either to a solution of a problem of extremal value or to problems depending on that. An example: if we have a surface, two points A and B on it, and a rectifiable arc joining A and B, then there is a shortest arc joining A and B.

After that introductory part to the thesis there followed three chapters with "applications" of that extended principle, namely to the problem of tight packing the space by congruent bodies (chapter I) and to some differential equations (chapters II and III). All three chapters contained new, original results, in particular (chapter III) concerning Holmgren's differential equation

d2y / dx2 + λ f(x,y) = 0.

The faculty, including Hilbert himself, was enthusiastic and granted to Steinhaus Ph. D.summa cum laude.

Steinhaus decided early to become a professional mathematician but for the time being he was only a "private scientist", as he liked to call himself. The three years 1911-1914 Steinhaus has spent between Jasło and Cracow, playing tennis, rowing, and making some mathematics. There also was a short military episode: enlisted into Austrian army,Steinhaus was terror-stricken by the drill he met there but managed, with the help of his mother, to be dismissed.

In the years 1911-1914Steinhaus wrote 2 papers on a generalisation of the concept of limit, 6 on trigonometric series, and 2 on potential series. The problem of generalisation of the limit concept consists in finding a procedure which applies to all number sequences, does not change limits of convergent ones and provides limits to some divergent ones. Steinhaus offered comments and proposed an axiomatics to a set of sequences possessing such a limit[3].

His first paper on trigonometric series provided an astonishing construction of a trigonometric series,

n=0 (ancos n + bnsin n) ,

the coefficients of which tend to 0 but which is divergent for every  [4]. Next papers in this group reveal his life long fascination with trigonometric and potential series which will find later its culmination in the joint monograph [5].The monographhas become the standard reference book up to 1960s.

A promising beginning was disrupted by the outbreak of World War I.The family found a temporary refuge in Vienna where young Steinhaus volunteered to Polish troops within Austrian army, called Legionyand headed by Piłsudski. As a soldier in the first artillery regiment he took a part in the Volhynia campaign 1915 against Russian army. His cousin was mortally wounded there and died shortly thereafter but Steinhaus survived. After the campaign he managed to be relieved and became a clerk in the Centre for Economic Reconstruction of the Country. He was honest to admit that he was a poor clerk, being interested in something else.

In that sad time of war, bringing misery and ruins, there came, however, two events which positively influenced his later mathematical life: meeting Banach and an invitation to Lvov.

One day in1916 Steinhaus took a walkin Planty, a park surrounding the old city of Cracow. There he heard two young men talking about the "Lebesgue measure", a new and not widely known concept in those days. It was so unexpected that Steinhaus turned around, came to the bank and introduced himself. The two men were Stefan Banach and Otton Nikodym, and soon a group comprising them and some other mathematicians living then in Cracow (including Leon Chwistek, Włodzimierz Stożek, Władysław Ślebodziński, Witold Wilkosz and some others) begun to meet on a regular basis.

Steinhaus offered to Banach a problem: does there exist an integrable function, the development of which into trigonometric series is not convergent in the first arithmetic mean? In a few days Banach returned with a construction of such a function. The idea was correct but the proof had some gaps. Steinhaus filled them up and this was the first Banach's paper, joint with Steinhaus [6]. The paper appeared only in 1918 but war timesare not friendly to pure mathematics.

The second fateful event was a meeting with professor Józef Puzyna who headed a mathematical chair at the Lvov university. Learning about Steinhaus and his mathematical achievements, he had persuaded him to make a habilitation in Lvov. For Steinhausit was like a godsend. The procedures were burdensome (as they still are) but Steinhaus passed them successfully in 1917. In that way he received venia legendi, that is, the right to teach at the Lvov university. He was happy. After succeeding to get a transfer from the main office of his Centre in Cracow to its branch in Lvov, he resettled there and started to lecture. It was 12 years after he entered the university as a student. And soon afterwards he invited Banach to join him in Lvov, securing him the position of an assistant to Prof. Antoni Łomnicki at the Polytechnic.[7]

In the years 1919-1920, which thus marked the end of Steinhaus' youth and the beginning of his academic career in Lvov, he published, besides another 6 papers on trigonometric and potential series, 2 papers in his new areas of interest.

In 1919 functional analysis was not yet recognized as a separate field of mathematics (even the name did not yet exist) but several function spaces have been already studied. In his paper in that new field Steinhaus considered the space of Lebesgue integrable functions and found a canonical form of any continuous and additive functional in that space [8]. It was the very first paper of a Polish mathematician in that field, and it found its place in the history of functional analysis [9].

Steinhaus always readily shared his ideas with friendly listeners, and so it seemscertain(it is, however, my guess) that Banach's interest in functional analysis was due to Steinhaus. The first fruits of that interest will come to light in Banach's Ph.D. thesis in 1920 which became a strong stimulus to the development of functional analysis as a whole and led the foundation to the theory of Banach spaces [10].

The second Steinhaus' paper concerning a new topic was a note on measurable sets [11]. Steinhaus proved, referring to earlier results of Sierpiński, that the set of distances in a linear set of positive measure contains an interval [0,) for some  > 0. Measure theory will later become the field of intensive research in Lvov by Steinhaus himself (particularly in a relation to probability theory which he intended to build upon measure theory) and by others including Stefan Banach, Alfred Tarski, Stanisław Ulam.

The two events - meeting Banach and influencing his interest in functional analysis, and coming to Lvov to become eventually Puzyna's successor after his untimely death in 1919 - have shaped Steinhaus' life. His formation as a mathematician has been completed and his further career has been firmly established. He became professor at the Lvov university in 1920 (ordinary professor in 1923) and was, together with Banach, one of thetwo founders and leaders of the Lvov school of mathematics which flourished during the next two decades.[12]

Until 1920 Steinhaus published 24 papers (of which it was shortly reported above), thus revealing main features of his mathematical talent. He liked novelties, read extensively, and in his papers he always referred to some very last discoveries, often improving them and/or generalising. Following a certain classification of scientific talents he may be compared to a butterfly, that is, he easily took from one topic to another, offering ingenious insights, but not particularly interested in their further development. His Ph. D. thesis was, essentially, on differential equations but with a strong ingredient of new set-theoretical ideas.Trigonometric series fascinated him mainly as a tool to provide counter-intuitive examples (he added many new ones), again often referring to recent papers by Lusin, Sierpiński and others.He early recognized functional analysis as a promisive independent field, contributed to it (also later on) and made it, together with Banach, a cornerstone of the Lvov school. However, hesoon lost interest in it.In measure theory he recognized the proper foundation of probability theory(which was then badly needed) and always promoted that point of view, getting a "semi-final" of its axiomatization [13], but later on developed a different approach and considered probability as an essential tool to study nature and society. Later years will bring another examples of the diversity of his mind, to mention only his early contribution to game theory, his brilliant popular book on mathematics which he saw everywhere [14], and applications of mathematics of which he became master in Wrocław.

On Steinhaus' tombstone one can read his creed: Between matter and spirit there is mathematics.

1

[1]The lecture was read at the conference organized by Hugo Steinhaus Center of Wrocław Polytechnic to celebrate centenary of his Ph.D. thesis.

[2] H. Steinhaus, Neue Anwendungen des Dirichlet'schen Prinzips, Göttingen 1911; reprint: H. Steinhaus, Selected Papers, Warszawa 1985, pp. 47-87.

[3] H. Steinhaus, Der Begriff der Grenze, Math. Ann. 71 (1912), pp. 88-96; reprint: H. Steinhaus, Selected ... , op. cit., pp. 101-108.

[4] H. Steinhaus, O pewnym szeregu trygonometrycznym rozbieżnym, C.R. Soc. Sci. Lettr. de Varsovie, 5 (1912); reprin: H. Steinhaus, On a certain divergent series, in:H. Steinhaus, Selected ..., op. cit., pp. 109-112.

[5] S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen, Monografie Matematyczne 6, Warszawa 1936.

[6]S. Banach, H. Steinhaus, Sur la convergence en moyenne de séries de Fourier, Bull. Intern. Acad. Sci. Cracovie, Cl. Sci. Math. Nat., Année 1918, pp. 87-96; reprint: S. Banach, Oeuvres II, Warszawa 1979, pp. 365-374; reprint: H. Steinhaus, Selected ..., op. cit., pp. 363-372.

[7]Banach studied at the Lvov Polytechnic in the years 1911-1914, but after fleeing in 1914 before Russian offensive to his native Cracow he never completed his studies. And in Cracow he met Steinhaus to become eventually a mathematician.

[8]H. Steinhaus, Additive and stetige Funktionaloperationen, Math. Z. 5(1919), pp. 186-221; reprint: H. Steinhaus, Selected ..., op. cit., pp. 252-288.

[9] J. Dieudonné, History of Functional Analysis, Amsterdam 1981 (p. 128).

[10] The thesis has been presented in 1920 and the degree was then granted, but the paper appeared in print two years later: S. Banach, Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math. 3 (1922), pp. 133-181; reprint: S. Banach, Oeuvres II, op. cit., pp. 305-348. For details of the discovery see R. Duda, The discovery of Banach spaces, in: W. Więsław (ed.), European Mathematics in the Last Centuries, Conf. Będlewo 2004, Stefan Banach International Mathematical Centre and the Institute of Mathematics of WrocławUniversity, 2005, pp. 37-46. See also: R. Duda, On the origins of Functional Analysis and the LvovSchool of Mathematics, Comment. Math., Tomus specialis in honorem Iuliani Musielak, 2004, pp. 5-45; A. Pietsch, History of Banach Spaces and Linear Operators, Birkhäuser 2007.

[11] H. Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fund. Math. 1 (1920), pp. 93-103; reprint: H. Steinhaus, Selected ..., pp. 296-304.

[12] R. Duda, Lwowska szkoła matematyczna [The Lvov School of Mathematics], Wrocław 2007. It is a book in Polish, but there are shorter aticles: R. Duda, Die Lemberger Mathematikerschule, Jber. DMV, 112.1 (2010), pp. 3-24; R. Duda, The Lvov School of Mathematics, Newsletter EMS, issue 89 (December 2010), pp. 40-50.

[13] K. Urbanik, Idee Hugona Steinhausa w teorii prawdopodobieństwa [Ideas of Hugo Steinhaus in the theory of probability], Wiadom.Mat. 17 91973), pp. 39-50; H.-J. Girlich, Łomnicki-Steinhaus-Kolmogorov: Steps to a modern probability theory, W. Więsław (ed.), European ..., op. cit., pp. 47-56.

[14] H. Steinhaus, Kalejdoskop matematyczny, Lwów 1938 (followed by 3 next editions); English translation:Mathematical Snapshots, 1938 (followed by 3 editions); also translated into Bulgarian, Czech, French, German, Hungarian, Japan, Roumanian.