Up to the end of the 19th century Poland was not noted for mathematics. The mathematical results ...

Krzysztof Ciesielski, Zdzislaw Pogoda (Jagiellonian University, Kraków, Poland)

 

 

Stefan Banach and the Lvov Mathematical School

 

 

This article is also available in Polish as a pdf (541 kB).

Up to the end of the 19th century Poland was not noted for mathematics. The mathematical results obtained by Jan Sniadecki, Jan Brozek, Jan Kochanski and some others were not enough to make Polish mathematics famous throughout the world. Note that in the 19th century, when mathematics in the world developed enormously, in times of Carl F. Gauss, Augustin L. Cauchy and Bernhard Riemann, Poland did not exist as an independent country.

 

Poland had for two centuries been divided among Germany, Russia and Austria. Only in the area under Austria (which included Cracow and Lvov) the Poles had some opportunities of scientific research. Nevertheless, up to the First World War the best Polish scientists, like Maria Sklodowska-Curie, obtained their results mainly abroad. Some mathematicians, like Józef Hoene-Wronski, Franciszek Mertens and Stanislaw Zaremba also worked abroad. Nevertheless, their advanced results did not make Polish mathematics in the end of the 19th century famous enough.

Everything changed after the First World War. Poland became a world mathematical power. In 1900 Zaremba moved to Cracow and created there a mathematical centre. Immediately after the war some Polish mathematicians formed the Warsaw Mathematical School. Their research were connected mainly with topology, a branch of mathematics which began to flourish at that time. The names of Polish mathematicians were widely known throughout the world. However, the greatest role in the fame of Polish mathematics was due to the Lvov Mathematical School and their members, among them, above all, Stefan Banach.

Hugo Steinhaus used to say that the discovery of Banach was his greatest mathematical discovery. This is a sentence of a particular importance, as Steinhaus was one of the greatest Polish mathematicians, the author of many outstanding results, one of the creators of the Polish Mathematical School. How was Banach ”discovered” by Steinhaus?

In 1916, Steinhaus, by then a well-known mathematician, was taking a walk. Suddenly he heard the words ”Lebesgue integral”. Today this is one of the basic terms in mathematical analysis, but at the time it was a recent discovery known almost exclusively to specialists. Steinhaus was intrigued. He approached the two young people discussing mathematics – they turned out to be Stefan Banach and Otto Nikodym (later a well-known mathematician) – and joined their conversation. At one point he told them about a problem he had been working on for some time.

Great was his surprise when next day Banach brought him the solution. Stefan Banach was born out of wedlock in Cracow in 1892. His mother was Katarzyna Banach and his father was Stefan Greczek. He was brought up by the owner of a laundry, to whose care he was entrusted from birth. After completing school, Banach decided to study engineering in Lvov. He liked mathematics but decided that very little could be added to such a developed subject. Study at the Technical University did not appeal to him; he definitely preferred far-reaching generalizations to the problems he encountered there. He had to tutor in order to support himself. Small wonder that it took him four years to earn his so-called half-diploma (which required completion of two years of course work).

At the outbreak of the Second World War Banach returned to Cracow and enriched his mathematical knowledge by independent study. In mathematics, he was self-taught. He read a great deal and sporadically attended lectures at the Jagiellonian University. He also engaged in many discussions with his friends Witold Wilkosz and Otto Nikodym. Steinhaus realized that Banach had a superb mathematical talent. Through his intercession, Banach, who had not earned any degree, was appointed to an assistantship at the Lvov Technical University.

In 1920 Banach was granted the doctoral degree by the Jan Kazimierz University at Lvov. Banach did not complete any course of study. Soon thereafter he was appointed professor. The story of the curious way Banach obtained his doctoral degree was told by Andrzej Turowicz, a priest and a Benedictine friar, who graduated in mathematics from the Jagiellonian University in 1928 and lectured at the Lvov Technical University before the Second World War. When he started working at the university, Banach had to his credit many important results and kept on making new mathematical discoveries all the time. When told that it was time for him to present a doctoral dissertation, Banach replied that he would do so as soon as he discovered a result better than the ones he had found so far.

Finally the university authorities lost patience. Somebody wrote down Banach’s remarks on some problems and these was accepted as a splendid doctoral dissertation. But an exam was also required. One day Banach was accosted in the corridor by a colleague and asked to go to the dean’s office, because ”somebody came and wants to know some mathematical details, and you will certainly be able to answer his questions”. Banach willingly answered the questions not realizing that he was being examined by a commission that had come to Lvov for this purpose. Such a procedure would probably be impossible today.

Those who knew Banach claimed that he attached little significance to nonmathematical matters. He talked and thought mathematics all the time. He was always full of new ideas. Only a small part of his ideas and results was written down. This was because he thought it more interesting and important to do research than to write up his findings. It was said that in order to make sure that posterity would get all of his results it would have been necessary for him to be followed by three secretaries who would take down all he said. Banach was very pleasant and friendly. He wasn’t conceited, he didn’t make an impression of being a great man. Although, as Turowicz said, one would guess that he knew what to think about himself.

An important role in shaping the atmosphere the Lvov mathematics worked in was played by their get-togethers in the ”Scottish Café” on Academic Street, near the university. A great many of them, including Banach, spent long hours at the café eating, drinking, and posing and solving mathematical problems. They had the habit of writing solutions on marble table-tops. In this way many a theorem was lost forever. Finally, Banach’s wife bought a notebook in which the mathematical habitués of the café recorded their problems. The notebook called the ”Scottish Book” remained in the café and was brought by a waiter at the request of mathematical patrons.

Sometimes the person who posed a problem offered an award for its solution. Some of the rewards were unusual. Thus Stanislaw Mazur offered a live goose for the solution of one of his problems. This was in 1936. A ”mere” 36 years later the problem was solved by the then 28-year-old Swedish mathematician Per Enflo, who later came to Warsaw and received the prize from Mazur. Some of the café sessions lasted many hours. In fact, one of them lasted 17 hours and yielded an interesting result that was wiped off the table-top by a conscientious waiter. There are those who claim that this was far from being the longest session, and that on one occasion two mathematicians got so involved in a discussion that they stayed in the café for 40 hours! Many anecdotes, legends, and tales are associated with the café sessions.

Once Stanislaw Mazur presented a problem, and Herman Auerbach started to thinking it over. After some minutes Mazur, wanting to make a puzzle more interesting, added that he offered a bottle of wine as a reward. After a while, Auerbach said: ”Oh, I give up. Wine does not agree with me”. The Scottish Café in 1978 Another story involves Henri Lebesgue, who arrived in Lvov in 1938 for the award of an honorary doctorate by the university. Lebesgue gave two lectures and, of course, was invited to the Scottish Café. A waiter gave Lebesgue the menu. Lebesgue, who did’t know Polish, studied the menu for a while, gave it back and said:” I eat only dishes which are well defined”.

The frequent café visits reflected certain aspects of Banach’s personality and character. When he wasn’t busy lecturing he could almost certainly be found in the Scottish Café. Its noise and airlessness seemed to suit him perfectly. There he could talk endlessly about mathematics, solve problems, and pose new ones. As a rule, after a long mathematical sessions at the Caf´e, he would arrive the next day with sketches of the solutions of most of the posed problems. Banach was surely the greatest mathematician in Lvov group, but there were also other excellent scientists.

The ”discoverer” of Banach, Hugo Steinhaus (1887–1972) was an exceptional person. He was remarkably many-sided; he obtained significant results in many different areas of mathematics. A significant part of his scientific work involves practical, sometimes very surprising, applications of mathematics. He had very original ideas and knew a lot. He was a man of great culture and deep knowledge of literature.

His aphorisms, remarks, and thoughts are famous to this day. Here are a few that are (unfortunately, most of them, including the best, are not translatable): ”It is easy to remove God from his place in the universe. But such good positions don’t remain vacant for long”. ”Strip-tease should be strictly forbidden. This is the only way of keeping this beautiful and useful custom alive.” ”It is easy to go from the house of reality to the forest of mathematics, but only few know how to go back.” Once, when somebody was decorated with a medal, Steinhaus said: ”Now I know what to do in order to be awarded a medal. Nothing, but for a very long time”. After the Second World War, Steinhaus was elected a member of sone scientific committee. Among the members of this committee there were many poor scientists who were the members of it because of political reasons.

Once Steinhaus did not come for the meeting of the committee and he was asked to explain the reason of his absence. He answered: ”I am not going to give any reason of my absence until some others will give reasons for their presence here”. He used to say that ”a computer is an extremely efficient idiot”. He was an accomplished populariser of mathematics. His book ”Mathematical Snapshots” first published in 1938, was translated into many languages. Strangely enough, it was not reissued in Poland between 1957 and 1990! Stanislaw Mazur (1905–1981), a mathematician who offered a goose as a reward for solving one problem, Banach’s pupil and friend, was also an excellent mathematician. Like Banach, he did not publish many of his results, but for another reason.

Banach had too many ideas and results, whereas Mazur did not like publishing. For Mazur only two things were interesting: mathematics and communism. Before the war it was not known that he was a member of the Communist Party. Mazur was 13 years younger than Banach. He started his study when Banach was already a professor. Nevertheless, Banach treated Mazur as a partner. They worked together. Mazur frequently judged Banach’s ideas, gave details of proofs.

Other Banach’s excellent pupils working in Lvov were Juliusz Pawel Schauder and Wladyslaw Orlicz. In the thirties, Kazimierz Kuratowski, a Warsaw mathematician, came for some years to Lvov. One of Kuratowski’s pupils was Ulam. Stanislaw Ulam (known later throughout the world as Stan Ulam) was 5 born in 1909 in Lvov, where he studied and initially worked. Already as a first-year-student he obtained original mathematical results which were soon published. Following an invitation by John von Neumann, one of the greatest mathematicians of the first half of the 20th century, he went to the United States in 1935 and settled there. Among other things, Ulam is most famous for his research in nuclear physics, performed at Los Alamos for 25 years (1943-1967). He was one of the discoverers of the theoretical foundations of the construction of the hydrogen bomb. He had broke scientific interests and obtained important results in various areas of mathematics (set theory, topology, measure theory, group theory, functional analysis, ergodic theory, probability, and game theory) as well as in a number of sciences (technology, computer science, physics, astronomy, and biology). He developed original methods of propulsion of vessels moving above the earth’s atmosphere. Ulam died in 1984. His life and work are described in the book ”From Cardinals to Chaos”, published posthumously at Los Alamos, and in his autobiography ”Adventures of a Mathematician”.

Another young mathematician, who moved abroad from Lvov before the war, was Mark Kac, well known because of his achievements in probability theory and statistics. The names of many mathematicians from the Lvov Mathematical School are until now very well known in the world, but for sure the most frequently mentioned is Banach’s name. It turned out that Banach is the mathematician (from the whole world) who is mentioned most frequently in the titles of mathematical scientific papers in the 20th century! Banach’s name is associated with many important, by now classical theorems: the Hahn-Banach Theorem, the Banach-Steinhaus Theorem, the Banach Open Mapping Theorem, the Banach-Alaoglu Theorem, the Banach Closed Graph Theorem, and the Banach Fixed Point Theorem.

And above all, there is the fundamental mathematical concept of a Banach space. What is it? We learn in school about straight lines, planes, and three-dimensional space. We can describe these geometric objects by means of numbers. Specifically, we can identify the points on a straight line with single real numbers, the points on a plane with pairs of real numbers, and the points in space with triples of real numbers. This idea can be extended to the study of finite sequences of numbers. These sequences can be added, and multiplied by numbers, very much like vectors. In this way we create so-called finitedimensional spaces.

We can – and do – go further. We add, and multiply by constants, numerical-valued functions, by defining the sum of two such 6 functions at a point to be the sum of their values at the point in question. In this we are no longer dealing with a finite-dimensional spaces. It turned out for a variety of reasons that function spaces are very useful in many investigations and applications. To a large extent, modern mathematics is concerned with the study of general structures, specific models of which have been known for a long time. One advantage of studying a general structure is the economy of thought: a theorem proved for the general structure need not be re-proved for its different models. Moreover, the general proof makes it easier to identify the properties utilized in the course of the proof and thus makes it more transparent.

It is paradoxical but true that sometimes the general proof is easier than its particular versions. Moreover, it is frequently useful in unanticipated situations. But the essential thing is finding the right generalization. Insufficient generality can be too restrictive and a great deal of generality may result in a situation where little can be proved and applied. The space introduced by Banach attests to his genius; he hit the traditional nail on the head. A space whose elements can be added, and multiplied by numbers, is called a vector space and its elements are called vectors. From the viewpoint of mathematical analysis and its various offshoots, vector spaces (without any additional structure on them) are of relatively little interest.

At the beginning of the 20th century one of the greatest mathematicians of the world, David Hilbert, introduced a kind of vector space – now known as Hilbert space – in which one could define perpendicularity. Notwithstanding its tremendous importance and its many applications, the notion of a Hilbert space turned out to be too restrictive for some very significant purposes. In the early phase of the study of Hilbert space mathematicians introduced in it the notion of a norm, which corresponds, roughly, to the notion of the length of a vector anchored at the origin. However, for some problems the concept turned out to be too general. At this point Banach came up with what turned out to be the ideal structure, namely, the notion of a normed vector space which has the additional property of so-called completness.

Roughly speaking, completness means, that if the distance between the elements of an arbitrary sequence tends to zero, then this sequence must have the limit (the elements of the sequence ”go somewhere”; in formal language – every sequence satisfying the Cauchy condition is convergent). A straight line, a plane, a three-dimensional space are the simplest examples of Banach spaces. Generally, more complicated spaces are considered. Some spaces of functions are of particular interest. It is safe to say that by 7 singling out the class of complete normed vector space Banach ”hit the jackpot”. It turned out that the property of completness was used in an essential way in proving many important theorems.

Banach’s great merit was that, in principle, it was thanks to him that the ”geometric” way of looking at spaces was initiated. The elements of some general spaces might be functions or number sequences, but when fitted into the structure of a Banach space they were regarded as ”points”, as the elements of a ”space”. At times this resulted in remarkable simplification. The great merit of Banach space is that, in spite of their abstractions and great generality, they have properties that accord with many intuitive notions associated with the geometries of the plane and of the three-dimensional space.

Today, eighty–odd years after its introduction, the notion of a Banach space remains fundamental in many areas of mathematics. The theory of Banach spaces is being developed to this day, and new, interesting, and occasionally surprising results are obtained be many researches. In particular, some really important results were obtained recently by William Timothy Gowers. Some problems he solved waited for the solution since Banach’s times. For his research, Gowers was awarded in 1998 with Fields medal, the most important reward in mathematical world. Moreover, there are many problems related to Banach spaces waiting for a solution.

The name ”Banach space” was probably used for the first time by Maurice Frechet, in 1928. The Lvov mathematician quickly showed the usefulness of the concept by proving in remarkably simple ways many difficult theorems which generalized certain, seemingly even more difficult, special cases. It should be pointed out that the eminent American mathematician Norbert Wiener arrived independently at the idea of Banach space (for a time one spoke of Banach-Wiener spaces), but decided that the relevant axioms implied excessive generality and were impractical from the viewpoint of applications. A few years later, after seeing the splendid uses of Banach spaces, he changed his mind and admitted an error of judgment.

Banach and his collaborators made an important contribution to the emergence of the vital area of mathematics known as functional analysis. Functional analysis can be described in a rough and imprecise manner as the study of the properties of certain functions whose domains are various Banach spaces. Functional analysis makes it possible to solve many problems that belong to other areas of mathematics (in particular, problems related to the study of differential equations). The by now classical monograph on functional analysis is Banach’s ”Linear Operations”, published in Polish in 8 1931. A French translation, ”Théorie des opérations linéaires”, appeared in 1932.

There is an amusing story to the effect that, upon publication, Banach’s monograph was displayed in some Lvov bookshops on shelves labelled ”Medical Books”. The most famous Banach results concern functional analysis, but he obtained many outstanding results in many other branches of mathematics as well. In particular, he published papers with results on topology, real functions, measure theory, orthogonal series, set theory... Some problems stated in the ”Scottish Book” did not require advanced mathematics in its formulation. Particularly elementary there was a question stated by Stanislaw Ruziewicz: can one decompose a square into a finite numbers of squares no two the same size? Ruziewicz, who posed this problem in the book, said that he heard the problem from somebody from Cracow.

For several years mathematicians in Lvov tried in vain to solve this problem. At last, in 1938 R. Sprague, using an earlier observation by Zbigniew Moron, showed the construction of the dissection of the square into 55 different squares. Later on several dissections into smaller number of squares were shown. At last the case was definitively closed in 1978 by a Dutchman A.Duijvestijn who decomposed the square into 21 different squares (see the picture) and proved that it is impossible to decompose the square onto 20 or less different squares. An interesting theorem which can be formulated in elementary way is ”the sandwich theorem” proved by Banach. The problem was posed by Steinhaus. There are three pairwise disjoint solids (that is, two different solids have no 9 element in common) in the three-dimensional space. Can one halve all of them by means of a single cut? The answer is yes. A consequence is that it is possible to cut a sandwich with butter and ham into two parts each of which contains half of each of the three ingredients. Its proof is advanced and nontrivial.

The magnificent development of the Lvov Mathematical school was broke by the Second World War. In 1939 Lvov was captured by the Soviet Union. The new government immediately made efforts to destroy the Poles, Polish science and Polish culture in this area. Many people were imprisoned. About two millions of Poles from the east Poland were deported, mainly to Siberia and Kazakhstan. The communists were particularly interested in deporting the officers of the army, the lawyers, the writers and the scientists. Banach was not arrested and not deported. He was even allowed to continue his work at the university. Perhaps it was because that he was in fact really interested only in mathematics, perhaps because of the support of Mazur (as then his communist outlook on life became widely known), perhaps of the support of Soviet mathematicians he collaborated with. In 1941 Hitler’s soldiers took Lvov for 4 years.

Then Banach’s situation became worse. He was not longer allowed to lecture at the university. He lived in extremely difficult conditions. When the war finished and it was finally decided that Lvov would be a part of the Soviet Union, Banach planned to go to Cracow, where he would have taken a Chair of mathematics at the Jagiellonian University, but a few days before the removal, in 1945 he died, age 53. The story of the Lvov Mathematical School ended together with the Second World War. During the war, other excellent Polish mathematicians died. In Lvov, Nazists killed Juliusz Schauder and Herman Auerbach. Several others frequent guests of the Scottish Caf´e also did not survive. Also, Stanislaw Zaremba and Witold Wilkosz in Cracow died. A young very talented mathematician, Józef Marcinkiewicz from Vilnius, who spent some years in Lvov, was murdered together with other Polish army officers in Katyn by Stalin’s regiment.

Other excellent members of the Lvov School moved from there. Steinhaus came to Wroclaw (Dreslau) which because of the Jalta agreement was taken from Germany and included to Poland. Mazur was in Warsaw. He has got an important position because of his party connections, but he held this position for a very short time, because he never took the trouble to answer letters. He was interested in mathematics above all, so he continued his research. Orlicz 10 moved to Poznan. Andrzej Turowicz turned up at the monastery in Tyniec, close to Cracow, and became a priest and a monk. Otton Nikodym and, as was told earlier, Ulam and Kac emigrated to the United States before the war and if they even planned to go back to Poland, because of the war they definitively decided to stay there for good. Banach was buried at Lychakov Cemetery in Lvov.

Banach’s grave in Lvov Hugo Steinhaus said that the greatest Banach’s merit was that he definitively broke the opinion that Poles was not good in mathematics and science. Up to the beginning end of the 20th century Poland was not noted for mathematics and very short it became the world power in mathematics.