Home Page of Stefan Banach
# Home Page of Stefan Banach

"Stefan Banach"
by Hugo Steinhaus
(Wrocław)
[PDF]
STUDIA MATHEMATICA, SERIA SPECJALNA, Z. I. (1963) pp. 7-15
(posted on this web-site with the written permission of the Publisher)
Stefan Banach was born on 20 March 1892, in Cracow. His father, an
official in the railway administration, Greczek by name, came of a
peasant family who lived in the highlands. The particulars of
Banach's early childhood are unknown; however, we know that
immediately after his birth he was put under the guardianship of a
laundress, living in a garret in Grodzka Street (nr 70 or 71);
Banach was the name of her husband. Since that time Banach never
saw his mother, so that for all practical purposes he did not know
her. Neither did his father care for him. Since the age of 15,
Banach had to support himself by private teaching. He was very
keen on giving coaching lessons in mathematics. As far as
mathematics goes, he was self-taught. We do not know how or when
he acquired the knowledge of French but we do know that as a
schoolboy he read Tannery's "Introduction àla théorie des
fonctions". Before the First World War, he attended lectures
delivered at the Jagellonian University by S. Zaremba, but he did
so irregularly and only for a short time. A little later, he moved
to the Lwów Institute of Technology, and there he passed what was
called the "first examination" certifying that he had studied
engineering for two years. When the First World War broke out in
1914, Banach returned to Cracow. One summer evening, in 1916, as I
was walking along the "Planty"(1), I heard a conversation, or
rather only a few words. I was so struck by the words "the
Lebesgue integral" that I approached the bench on which the
speakers were sitting and, then and there, I made their
acquaintance. The speakers, Stefan Banach and Otto Nikodym, were
discussing mathematics. They told me they had another friend -
Wilkosz. It was not only mathematics that bound those three young
men together, it was also the hopeless situation of young people
in "fortress Cracow" (such was the official status of Cracow in
those days of the war), the insecurity of the future, the
difficulty of earning one's living, the lack of contacts not only
with foreign scientists, but even with the Polish ones - such was
the pervasive atmosphere of this city in 1916. But all that did
not prevent the three young men from spending a lot of time in
cafes discussing problems in mathematics amidst a noisy crowd.
Banach did not mind the noise; for some reasons, known only to
himself, he liked to sit quite near the orchestra.
I was studying the problem of the average convergence of Fourier
series and this was the very question that I put to him in 1916
when I met him in the "Planty". For some time I had been trying
to solve it myself. I was greatly surprised when, after a few
days, Banach brought me a negative answer with a reservation which
resulted from his ignorance of the Du Bois-Reymond example. Our
common note was presented by S. Zaremba to the Academy of Sciences
in 1917, to be published only in 1918. This delay was caused by
the war.
Banach dreamt of being appointed mathematics assistant at the Lwów
Institute of Technology. This dream came true in 1920, when Antoni
Łomnicki gave Banach the post. Now since he first arrived in Lwów,
Banach's position changed radically. His financial situation was
secure. He married and settled down in the University building
quarters on St. Nicolas Str. In 1922, his doctor's thesis appeared
in the 3rd volume of Fundamenta Mathematicae: "Sur les opérations
dans les ensembles abstraits et leur application aux équations
intégrates".
That was Banach's seventh paper, but the first to be dedicated to
the theory of linear operations. In the same year he was made an
Assistant Professor. When it came to Banach, the customary
University procedures were waived - he was awarded a doctor's
degree although he had never completed any formal course work. At
that time he was 30 years old. There was no lack of
acknowledgments from other sources either. In 1924 Banach became a
correspondent member of the Polish Academy of Sciences. In 1930 he
was awarded the Prize of the City of Lwów, and, in 1939, he was
given the Prize of the Academy. It is difficult to understand
nowadays why in that Academy there was no seat available for this
son of the streets of Cracow. But the Lwów mathematicians
understood at once that Banach would make Polish mathematics
famous. Before he came along there was, strictly speaking, no Lwów
school, because Sierpiński, soon after the First World War,
returned to Warsaw from which he had been driven away by that war,
and Janiszewski died soon afterwards. In the interwar period, the
Lwów school might have been characterized, first of all, by the
theory of operations, because its main achievements were in that
particular branch of mathematics. Banach took up linear
functionals, such as the integral and showed that the concept of
the integral may be widened so as to embrace all functions while
retaining the properties postulated by Lebesgue; although this
concept is ineffective, the proof of existence and the method of
carrying it out (Fund. Math. 1923) show Banach's power. His main
work is a book on linear operations. Published in 1932 as the
first volume of Mathematical Monographs (Warsaw, VII+254 pages) it
is nowadays well known in the whole world of mathematics, under
the title "Théorie des opérations linéaires". Its success is due
to the fact that owing to the so-called "Banach Spaces" it is
possible to solve in a general way many problems which formerly
called for special treatment and considerable ingenuity. There
were other mathematicians, some great, who had attempted before
Banach to build up a theory of operations. I remember what the
outstanding Göttingen mathematician, Edmund Landau, said of the
"Operazioni distributive" written by Pincherle; "Pincherle has
written a book without having proved any theorem" - it was quite
true. But there were also some more formidable competitors. This
is what Norbert Wiener, the author of cybernetics, writes in his
autobiography, "I Am a Mathematician", published in 1956. He
says that Fréchet, who was the first to give the form of a linear
functional in the L2 space could not make up his mind as regards a
system of postulates which would determine a general space such
that L2 would be only one of other numerous examples. Wiener takes
the credit for it himself. He tells us how Fréchet, whose guest
Wiener was in Strassburg (in 1920, on the occasion of the
Mathematical Congress) showed him Banach's article in "a Polish
mathematical journal". Fréchet was excited by the fact that
Banach had given, a few months earlier than Wiener, a system of
axioms of an infinitely dimensional vectorial space identical with
Wiener's system. "So" says Wiener, "the new theory was for some
time called: The Banach-Wiener space theory". "But", says
Wiener, "I wrote a few more papers on those problems, and
gradually gave it up" - "at present those spaces are justly
called after Banach's name alone... "(2). After this statement,
Wiener dedicated a few pages of his autobiography to that conflict
and explains why he left the battlefield; he thought that Banach's
theory was a formalism, which could not show to its credit a
sufficiently rich stock of original theorems hitherto unknown -
now he admits that he was wrong, for after 34 years which have
elapsed since the Strassburg Congress the Banach theory is still
popular as a tool of analysis and "only now begins to develop its
full effectiveness as a scientific method". Banach's fame reached
the United States even before the appearance of "Operations
lineaires". As early as 1934, in the Bulletin of the American
Mathematical Society (vol. 40, p. 13-16), J. D. Tamarkin wrote in
his review of Banach's book: "It presents a noteworth climax of a
long series of investigations started by Volterra, Fredholm,
Hilbert, Hadamard, Fréchet and Frederick Riesz, continued
effectively by Stefan Banach and his pupils". And then "The
theory of linear operations is in itself an exciting domain, but
its importance is enhanced by numerous and beautiful
applications". One of Banach's most gifted pupils, Stanisław
Ulam, writes thus in the obituary published in July, 1946, in the
Bulletin of the American Mathematical Society (v. 52, No. 7,
(1946), p. 600-633): "We have recently received the news that
Banach died in Europe soon after the end of the war. The great
interest aroused by his work is a well-known fact in our country.
Indeed, in one of the fields of his activity, in the theory of
infinitely dimensional linear spaces, the American school has
developed and is still supplying very important results. It is an
astounding coincidence of scientific intuition, which has
concentrated the efforts of numerous Polish and American
mathematicians in the same field..." And next; "Banach's work
has set off for the first time in a general case the success of
geometrical and algebraic approach to the problems of linear
analysis, reaching far beyond the rather formal discoveries of
Volterra, Hadamard and their successors. His results embraced more
general spaces than the works of such mathematicians as Hilbert,
E. Schmidt, von Neumann, F. Riesz and others. Many American
mathematicians, particularly the younger ones, have taken up the
idea of geometrical and algebraic study of linear functional
spaces, and this work is still (1946) going on vigorously and
bringing about important results".
I think that these opinions of outstanding scientists (one of whom
has played an essential part in the computation of the
thermo-nuclear hydrogen reaction) may suffice as a proof that
Banach knew how to take a leading role in the development of an
exceedingly important and new chapter of analysis, getting to the
fore of a group of eminent mathematicians, who had already tried
their forces in a similar trend.
May I be allowed to say for my part, as a witness of Banach's
work, that he was possessed of a lucidity of thought which
Kazimierz Bartel (3) called once "even unpleasant...". He would
never rely on a lucky stroke, he never expected that conjectures
desirable at the time would prove true; he often said that "hope
is the mother of fools". He adopted this contemptuous attitude to
optimism not only in mathematics, but also towards political
prophecies.
He was like Hilbert; he attacked problems directly - after
excluding all side routes through examples, he concentrated all
his forces on the only way left, taking a straight aim. He
believed that logical analysis of the problem, like a chess player
analyzes a difficult position, must bring him to a proof or to a
refutation of the theorem.
Banach's importance is not limited to what he achieved himself in
the theory of linear operations. On the list of his 68 works we
find studies written in collaboration with other mathematicians
and books concerning other domains. The paper on the decomposition
of sets into congruent parts, written together with Tarski (Fund.
Math. 6 (1924), p. 244-277), may be placed in both those classes.
It is a subject which reminds one of the school method of proving
Pythagoras' theorem by cutting a large square into parts of which
two small squares can be made; here, in three dimensions, the
result is unexpected: a sphere can be cut into several parts of
which two spheres can be made, each as big as the original one.
Personally I was greatly impressed by a short paper in the
Proceedings of the London Mathematical Society (vol. 21, p.
95-97). The problem consists in finding an orthogonal system
complete in L2 but incomplete in L. Banach chooses a function f(t)
which is (L), 01 f(t)dt = l, but such that 01 f2(t)dt = ; he
denotes by n(t) the sequence of all trigonometric functions
cos nt, sin nt and defines the numerical sequence cn by the
relation 01f(t) n(t) dt = cn; if we now 0 define the sequence
n(t) by n(t)= n(t) cn we shall of course have 01 f(t) n(t) dt =
0 for all n. It we orthogonalize and normalize the sequence
n(t), we shall obtain the required sequence n(t). The
cleverness of the proof lies in the fact that the auxiliary
sequence n(t) is deprived of the property which we demand from
the sequence sought. Banach's works on the convergences of
functionals are also known; they were originated by one of his
colleagues, generalized by Banach and given their final shape by
S. Saks (1927, Fund. Math. 9, p. 50-61). Banach took an interest
in the problem of complanation, i. e., the definition of the
concept of the area of curved surfaces. His definition, gave rise
to some investigations after the war (e. g., such as those being
conducted now by Prof. Kovanko in Lwów) - unfortunately no one
knows how to reproduce the essential lemma indispensable to show
the conformity of the Banach definition with the classical ones.
We regret to say that many valuable results of Banach and his
school's work were lost to the great detriment of Polish science
as a result of carelessness on the part of the school's members
and, first of all, on the part of Banach himself. Another of his
beautiful ideas is the replacing of the classical definition of
the oscillation of the function y = f(x) by one more fitting to
the epoch of Lebesgue, namely by the integral 0 L( )d , where L(
) denotes the number of intersections of the curve y = f (x) with
the straight line y = ; the readers may find it interesting to
know that this approach has a practical significance, e. g. it
permits a quick calculation in "dollar-days" of the bank credits
held up in factory storehouses in the form of raw materials
awaiting processing.
I do not want to say any more about the numerous and important
items on the list of the works of the originator of the Lwów
school and the founder of the journal "Studia Mathematica",
which have played a considerable part in the development of that
school and in the history of linear operations theory.
Let us revert to Banach's person and his immediate influence on
his environment. Banach was appointed full professor in 1927, but
neither before nor after that was he a professor in the solemn
sense of the word. His lectures were excellent; he never lost
himself in particulars, he never covered the blackboard with
numerous and complicated symbols. He did not care for verbal
perfection; all manner of personal polish was alien for him and,
throughout his life he retained, in his speech and manners, some
characteristics of a Cracow street urchin. He found it very
difficult to formulate his thoughts in writing. He used to write
his manuscripts on loose sheets torn out of a notebook; when it
was necessary to alter any parts of the text, he would simply cut
out the superfluous parts and stick underneath a piece of clean
paper, on which he would write the new version. Had it not been
for the aid of his friends and assistants, Banach's first studies
would have never got to any printing office. He hardly ever wrote
any letters and never answered questions addressed to him by post.
He did not relish any logic research although he understood it
perfectly. Neither was he attracted by any practical applications
of mathematics, although he could certainly go in for them if he
wanted to do so - a year after taking his doctor's degree he
lectured on mechanics at the Institute of Technology. He used to
say that mathematics is marked by specific beauty and can never be
reduced to any rigid deductive system, since, sooner or later, it
will burst any formal framework and create new principles. It was
not the utilitarian but the specific value of mathematical
theories that counted with him. His foreign competitors in the
theory of linear operations either dealt with spaces that were too
general, and that is why they either obtained only trivial
results, or assumed too much about those spaces, which restricted
the extent of the applications to few and artificial examples -
Banach's genius reveals itself in finding the golden mean. This
ability of hitting the mark proves that Banach was born a
mathematician of the highest class.
Banach could work at all times and everywhere. He was not used to
comfort and he did not want any. A professor's earnings ought to
have supplied all his needs amply. But his love of spending his
life in cafes and a complete lack of bourgeois thrift and
regularity in everyday affairs made him incur debts, and, finally,
he found himself in a very difficult situation. In order to get
out of it he started writing textbooks. Thus the "Differential
and Integral Calculus" came to life in two volumes, the first of
which was edited, by the Ossolineum (1929, 294 pages) and the
second by the Książnica-Atlas (1930, 248 pages). This manual,
written in a concise and clear way, has enjoyed and is still
enjoying a great popularity with university students in the first
two years of their studies. The writing of secondary school
textbooks for arithmetic, algebra and geometry took up a lot of
Banach's time and effort. He wrote them in collaboration with
Sierpiński and Stożek. Some of them were written by himself. He
never copied any of the existing textbooks. Thanks to his coaching
experience, Banach realized very well that every definition, every
deduction and every exercise is a problem for the author of a
school-book who cares for its didactic value. In my opinion Banach
lacked only one of the many talents that an author of schoolbooks
needs: spacial imagination. Banach's "Mechanics for Academic
Schools" (Mathematical Monographs 8, 9) is the fruit of the
experience collected during his numerous lectures on mechanics at
the Institute of Technology. This two-book course, published in
1938, was issued once more in 1947, and, a few years ago, it was
translated into English.
In order to give an account of Banach's importance for science in
general, and for Polish science in particular, we should mention
the names of his immediate followers: Mazur and Orlicz are his
direct pupils; they represent the theory of operations in Poland.
Their names to be seen today on the cover of "Studia
Mathematica", signify the direct continuation of the Banach
scientific program which found an expression in this journal.
Stanisław Ulam (who owes his mathematical initiation to
Kuratowski) after taking a doctor's degree also entered Banach's
orbit. Banach with Mazur and Ulam formed the most important corner
at the Scottish Café in Lwów. That was the place of gatherings to
which Ulam alludes in the already quoted obituary: "it was hard
to outlast or outdrink Banach during those sessions". There was
even a session which lasted 17 hours - its result was the proof of
a certain important theorem pertaining to Banach Spaces - but
nobody took it down at the time and nobody can reproduce it
today... Probably the top of the table covered with pencil marks
was washed as usual by the Café help. Such was the lot of a good
many theorems proved by Banach and his followers. It was,
therefore, a great merit on the part of Mrs. Łucja Banach - who is
already lying at rest in the Wrocław cemetery - to buy a thick
notebook with a stiff cover and to entrust it to the headwaiter of
the Scottish Café. The problems were then entered on the first
pages of the successive leaves, so that the answers, if and when
they were found, could be entered on the empty pages next to the
text of the questions. This original "Scottish book" was at the
disposal of every mathematician who asked for it in the Café. In
some cases prizes were promised for solutions - they ranged from a
small cup of black coffee to a live goose. Nowadays, he who smiles
condenscendingly on hearing about such ways of cultivating
mathematics should try to understand that, according to Hilbert's
opinion, the formulation of a problem is half the solution, and
the list of unsolved and proclaimed problems makes people seek for
a solution and is a challenge to all those who are not afraid to
attempt difficult tasks. This state of mental readiness creates a
scientific atmosphere. Among those of Banach's students who fell
at the hands of murderers wearing uniforms adorned with the
swastika, the most outstanding was undoubtedly P. J. Schauder, the
laureate of the international Metaxas Prize, which was awarded to
him and Léray ex aequo. It was Schauder who noticed how important
were the Banach spaces for the boundary problems of partial
differential equations. The difficulty lay in the selection of
proper norms; it was overcome by Schauder and, thanks to that
scientist, young at that time, the wreath of victory in such a
classical theory as partial differential equations was shared by
France and Poland.
The later history of Banach's life passed under the shadow of the
Second World War. In 1939-1941 he was dean of the Faculty in Lwów
University, and even a correspondent member of the Kieff Academy,
but after the German invasion (at the end of June 1941) he had to
feed lice in Professor Weigel's Bacteriological Institute. He
spent several weeks in prison, because some people engaged in
smuggling German marks were found at his home. By the time the
case was cleared, Banach succeeded in proving a new theorem (4).
Banach was, first of all, a mathematician. He did not take much
interest in politics, although he had a shrewd approach to every
situation in which he happened to find himself. Nature did not
impress him at all. Fine arts, literature, the theatre were to him
second-rate amusements, which could, at their best, fill up the
few short intervals in his work - on the other hand, he enjoyed
jovial company and a drink. That is why the concentration of all
his mental energy in one direction found no impediment. He did not
like to delude himself and he knew very well that there is only a
very small percentage of people who can understand mathematics.
One day he said to me, "I'll tell you something, old chap!
Humanities are more important in secondary schools than
mathematics - mathematics is too sharp an instrument, it is not
made for children to play with..."
It would be wrong to think that Banach was a dreamer, a
self-denying ascetic or apostle. He was a realist who even
physically did not look like a candidate for a saint or even for a
Tartuffe. I do not know whether there still exists, but there
certainly did exist 30 years ago, the ideal of a Polish scientist,
based not on the observation of reality but on the spiritual needs
of the epoch which found its expression in Stefan Żeromski's
works. Such a scientist was supposed to work, far from all worldly
pleasures, for a rather indefinite society, which forgave him a
priori the ineffectiveness of his work, taking no heed of the fact
that in other countries the greatness of scientists was gauged not
by the extent of personal self-denial, but by what they had done
for science. In the interwar period the Polish intelligentsia was
still under the suggestion of this self-mortification ideal, but
Banach never submitted to it. He was healthy and strong, his
realism was almost cynical, but he gave to Polish science, and
particularly to Polish mathematics, more than anyone else. Nobody
helped more than he to dispel the harmful opinion that, in
scientific competition, lack of genius (or at least talent) can be
compensated by other qualities, rather difficult to define. Banach
was aware of his value; he realized fully what he was
accomplishing for science. He stressed his highland descent, and
his attitude to sophisticated members of the "intelligentsia
without portfolio" was a contemptuous one. He lived to see the
German collapse in Lwów, but a little later, on 31 August 1945 he
died. He was buried at the cost of the Ukrainian Republic. One of
the streets in Wrocław was named after him (5). His collective
works will be published by the Polish Academy of Sciences. His
greatest merit is the overthrow and final annihilation of the
Polish inferiority complex with regard to science, a complex
camouflaged by the exaltation of mediocre thinkers. Banach never
suffered from that complex - in his mind the spark of genius was
combined with an astounding inward spirit, which told him again
and again the words of Verlaine: "Il n'y a que la gloire ardente
du métier !"(6) - and mathematicians know very well that their
craft consists in the same mystery as the craft of the poets.
FOOTNOTES
(l) A park surrounding the city.
(2) This name was introduced by Fréchet.
(3) K. Bartel, Professor of the Lwów Institute of Technology,
prime minister, murdered by the Gestapo in 1941.
(4) Recently. I met in Chicago one of my former students. She told
me that she had obtained her doctor's degree from Banach; this
proves his participation in the underground teaching in those
days.
(5) Recently in Warsaw too.
(6) "There is only one thing: the ardent glory of the craft!"
We deeply thank Prof. dr. hab. Stanisław Janeczko, the Head
of the Mathematical Institute of Polish Academy of Sciences, for
his permission to post copies of all articles about Stefan Banach
published in "Studia Mathematica" on this website.
Questions or comments about this page can be
sent to Emilia Jakimowicz
or Adam Miranowicz. We would also
appreciate every link from your pages to our Home Page of
Stefan Banach.

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On 13 Jan 2012, 18:52.