Juliusz P. Schauder

Center for Nonlinear Studies

Nicolaus Copernicus University in Torun

Chopina 12/18, 87-100 Toruń

e-mail: cbn@umk.pl

e-mail: cbn@umk.pl

Juliusz Schauder - Personal reminiscences

Topological Methods in Nonlinear Analysis

Journal of the Juliusz Schauder Center

Volume 2, 1993

**JULIUSZ SCHAUDER
- PERSONAL
REMINISCENCES **

R. S. INGARDEN

*Dedicated to the memory of my teacher, Juliusz Schauder*

Juliusz Schauder was my teacher of mathematics in secondary school and also at the University of Lwów where I studied theoretical physics. I knew him during a period of about 10 years, but his tragic death interrupted this acquaintance just at the moment when it seemed that it might have been the beginning of a period of fruitful collaboration. Schauder looked to me as a prospective collaborator on mathematical problems of physics and for this reason he invited me to his advanced seminar about continuous groups. In these notes I would like to say what I remember about my dear teacher, who gave me the first impulse in the direction of mathematical physics (an impulse which I followed without his guidance and not necessarily in his direction). It occurred only recently, and completely independently of the present paper, that I came back to Pontryagin's book, which we studied at the seminar, in order to use it directly in my current work.

Before I give my personal reminiscences I would like to recall briefly what is known about Schauder's life (cf. [1] and [2]). I will not write here about his work, since this would take us too much aside (for this cf., e.g., [3] and [4]; unfortunately, neither text is in English: one is in French and the other in Japanese !). I will begin by quoting his friend and collaborator, Jean Leray (cf. [5], p.11):

Cette oeuvre est en apparenee diverse: elle apporte des contributions souvent fondamentales a` quatre branches des mathematiques, qu'on a coutume d'etudier independemment l'une de l'autre. En fait, c'est ane oeuvre d'une grande continuite: Jales Schauder n'est pas an specialiste; il remonte a l'origine des problemes; il s'apercoit qu'ils exigent de tout antres procedes que ceux qu'on a pris l'habitude de leur appliquer; sa pensee progresse, sans souci des cloisonnements qui compartimentent arbitrairement la science et qu'ont du toujours bousculer les esprits vraiment originaux. Sa puissance intellectuelle s'allie aux plus belles qualites de caractere: il est un grand travailleur, tres scrupuleux. Il etudie "les problemes qui se posent et non ceux qu'on se pose"; son oeuvre n'est pas un jeu brillant, mais un rude labeur, qui finit par creer la methode dont l'elegante perfection orientera l'avenir.

As we shall see, all these observations conform very well to my experience of Juliusz Schauder.

PHOTO 1. Parents of Juliusz Pawel Schauder

Juliusz Pawel Schauder was born in Lwów on
September 25, 1899 as a son of Regina and Zygmunt Schauder. His father
was a lawyer in Lwów and also in Rohatyn where Juliusz attended
primary school. In 1917, at the regular age of 18, Juliusz finished secondary
school in Lwów and immediately afterwards was drafted into the Austrian
army. He was sent to the Italian front and fought there until he was captured
by the Italians as a prisoner-of-war. After the armistice 27,000 (The
great number of prisoners was a consequence of the battle by Vittorio Veneto
on October 24-30, 1918, and of the Italian actions afterwards, which led
to the armistice on this front. When Steinhans and Schauder were in Bologna
in 1929, they saw there an interesting monument. As Steinhaus writes, [2],
p. 108: "*In the middle of the town there is a fine town hall and
on it *a "bulletin of victory" engraved in stone. It gives
fantastic numbers of prisoners and guns taken in 1918 by Vittorio Veneto.
It is generally known that, after the Hungarian army turned for home, the
Austrian troops decided to do the same and laid down their arms to go home
through the Alps. But small Italian detachments in cars, pushing their
way through disordered crowds of soldiers, who had mostly long ago laid
down arms, barred the way and enrolled them into captivity completely unnecessarily.
At that time, the Italian staff was in Vienna." But these great numbers
turned out to be useful for the Polish cause. Together with Poles from
France and America the Polish Army in France became a considerable force
of 6 divisions. Afterwards, it helped Poland greatly in the fighting around
Poznan, Torun and Lwów.)^{ }of the Austrian
prisoners-of-war of Polish nationalily in Italy volunteered to join the
Polish Army organized in France, and among them was Juliusz Schauder. With
this army, the so-called Haller's Blue Army, he returned to Poland in 1919.
His military service ended the same year, and in the fall he started to
study mathematics and physics at the Jan Kazimierz University of Lwów.
It is characteristic of his approach that he should have chosen both mathematics
and physics, but he could not succeed in the study of both sciences because
of the situation of physics at the Lwów University at that time.
The university has in its history two theoretical physicists of the first
rank: Marian Smoluchowski (1872-1917) and Wojciech (Adalbert) Rubinowicz
(1889-1974). The first, however, left for Kraków in 1913, while
the other, although he came to Lwów in 1923, was then a professor
at Polytechnic. (He moved to the University only in 1937, and soon afterwards
I became his student). At the time of Schauder's studies the chair of theoretical
physics was for a few years temporarily occupied by an experimental physicist
Stanislaw Loria (1883-1958), who probably would not have attracted the
attention of the mathematically minded Schauder. (I
learned from K. Maurin that Wladyslaw Orlicz, who was a student of physics
at the University at approximately the same time as Schauder, once said
that when he went to Gottingen to prepare his Ph.D. in theoretical physics,
he realized that his education in Lwów had been absolutely insufficient.
He therefore resigned from becoming a physicist, turned, like Schauder,
to mathematics, and went back to Lwów.) Mathematics,
on the other hand, had just then started its period of most flourishing
development. Schauder's professors of mathematics were Stanislaw Ruziewicz,
Hugo Steinhaus, Eustachy Zylinski and, since 1922, Stefan Banach. Steinhaus
writes in his Memoires [2], p.99: "*Among the students,
many were soon found who clearly distinguished themselves by their abilities:
one such was Schauder, others were Orlicz, Halaunbrenner, Auerbach and
Mazur*".

In October 1923 Schauder obtained his Ph.D. in mathematics
on the basis of his first paper. It was published in 1926 [6]
in English (all his later papers were published in German or in French).
In 1925 he started working as a teacher of mathematics in a secondary school.
He taught first for two years in Przemyslany (60 km from Lwów),
and then (several years later) in Lwów up until the Second World
War. This was quite a heavy burden, but in spite of it he worked very intensively
in mathematics, and on January 17, 1928 he was able to make his "habilitation"
(*veniam legendi*) to become a'private docent' at the university;
the title did not mean money and so economically nothing had changed. Schauder
started, however, to give special lectures at the University and soon became
an influential member of the emerging (and later world famous) mathematical
school centered around S. Banach and H. Steinhaus. Other leading members
of the school were Stanislaw Mazur, Wladyslaw Orlicz, Stefan Kaczmarz,
and, somewhat after, Stanislaw Ulam and Marek Kac.(We
read in Steinhaus' Memoires [2], p.108: "*In the
fall of 1929 we went for a congress to Bologna. From Lwów came:
Antoni and Zbigniew Lomnicki, Stozek, Ruziewicz, Zylinski, Banach, Chwistek,
Kaczmarz, Nikliborc and Schauder. The congress was successful - there came,
among others, Hilbert, Weyl, Kellogg, Landau, Hadamard, Lichtenstein, de
la Vallee-Poussin, Serge Bernstein, Khinchin, Menshov, Lusternik and Haar*".)
Many years later Ulam, describing some special features of the mathematical
life in Lwów during the years before 1939, wrote that he remembered

*a mathematical session with Mazur and Banach lasting 17 hours without
in- terruption except for meals. ... Much of the mathematical work was
carried on in a way not usual in American mathematical centers. The mathematicians
in Lwów met not only in their classrooms and offices but spent long
hours every day in two coffee houses *(*Szkocka and Roma*)* which
served as informal meeting places. They discussed problems over coffee
or beer, and marble table tops and napkins took the place of blackboards.
It was hard to outlast or outdrink Banach during these sessions. He was
always there. The weekly *(*Saturday night!*)* meetings of the
mathematical society, where papers were presented, provided the more formal
discussions.* (cf. [5]).

Around 1935 Schauder obtained the position of a senior assistant under Professor Steinhaus. In 1937 the full staff of the Department of Mathematics of Lwów University was as follows ( cf. [7], p.77 ; as for a document, I give all the details in translation):

Department of Mathematics; 4 St. Nicholas Street, 2nd floor, tel. 211-79.

**Division A**

Head: Prof. Dr. Eustachy Zylinski | 11a Supinski St. |

Sen. Assistant: Doc. Dr. Herman Auerbach | 6 Konopnicka St. |

**Division B **

Head: Prof. Dr. Hugo Steinhaus | 14 Kadecka St. |

Sen. Assistant: Doc. Dr. Juliusz Schauder | 2 Zascianek St. |

Dep. Assistant Volunteer: Helena Plamitzer | 32 Gipsowa St. |

**Division C **

Head: Prof. Dr. Stefan Banach | 23 Zyblikiewicz St. |

Dep. Assistant M. A. Kazimierz Szalajko | 7 29th of November St. |

Sen. off. Messenger: Jozef Goral | 26 Mochnacki Str. |

As we see, we have here only three professors and two docents; all the other active Lwów mathematicians mentioned (or not mentioned) above were not included on the university payroll, (they belonged to Polytechnic or other schools). Although Schauder had the low position of a senior assistant, his inclusion in the above list was rather a distinction.

*** * * **

Let me now come to my direct personal reminiscences. It was probably in September of 1933 that I met Schauder for the first time. I remember this moment very well : Dr. Bednarowski, the director of our school (the III State Gymnasium in Lwów, named after King Stefan Batory), presented to the Class IV A, of which I was a pupil, a new teacher of mathematics, Dr. J. Schauder. I remember how he stood before the blackboard: he was a little bit stiff and shy, with his high forehead and intelligent, noble face. He was not tall, had blond hair, and his skin was a little yellowish and pale with small wrinkles, although he still looked young ( he was then 34, while I was 13). At first sight, his features looked slightly Semitic to me; this impression, however, disappeared rather soon. (I write about this impression since it happened to be so important for his fate later on). As an "A" class, ours was purely Catholic and Polish, only "B" classes had Jewish and Ukrainian students or those of other religious denominations. (The division was according to religion, not nationality, and was, I think, the unhappy idea of our director). I never noticed in our class anyone having reservations about the new teacher because of his Jewish extraction, although relations between nationalities and faiths in Poland and in Lwów were not, in general, always the best. But my classmates quickly found out that "Schauder" means in German "horror" ("zgroza" in Polish), so the new teacher obtained the nickname "Zgroza". He was usually kind-hearted and relaxed, but sometimes he would become a little nervous and stiff with bad students; so that in those rare moments when he spoke quickly and with some impatience, his nickname seemed to be appropriate. Obviously, we had no opportunity to learn of his higher scientific abilities from the elementary mathematics which he taught us. Our school was of the so-called "old classical" type, which means that it had Greek and Latin in its curriculum, but little mathematics and almost no physics. This type of 8th-grade gymnasium system was just then in liquidation; my class was the last of this system, while my younger brothers were already attending the new school system, without Greek but with more mathematics and physics. When I think now about Schauder's pedagogical abilities in secondary school. I cannot say that they were remarkable. We noticed the difference when comparing him with our previous teacher of mathematics, the old Roman Chobrzynski. A small group of scientifically minded colleagues to which I belonged, and which had founded, in the 2nd grade, a "Scientific Circle" ("Kólko Naukowe"), was very upset by Chobrzynski's retirement and death. He had had a spark of humor and we were enthusiastic about his exercises, which were based mostly on the old Austrian Kranz' book of problems in mathematics for secondary schools. From the time of Schauder's teaching I do not remember such emotions, although I do remember that his teaching was solid, systematic and very clear. I also remember the textbooks written by Banach, Sierpinski and Stozek; they were all excellent: clear, concise, and interesting. Especially remarkable was the book on geometry, the largest of all. Schauder used these books in teaching, although he did not follow them very closely.

My father was then also a teacher of mathematics and logic and a docent at Lwów university (a year later he became a professor of philosophy at the University and stopped working at a secondary school), so I know how hard a teacher's work was at that time. My father went to school every day at 8 o'clock in the morning and came back at about 1 in the afternoon, very exhausted. I under- stand therefore what a heavy burden it was for Schauder to teach while working intensively in mathematics during the afternoons and at nights. Schauder knew my father, and sometimes we met each other by chance on the street or in a park. I was always surprised at how relaxed and in what a good mood Schauder was after the strain of the school. In particular, I remember that one day my father took me to the Scottish Coffee House (my father used to play chess there with Herman Auerbach, an outstanding chess player) and we met Schauder there and talked to him for a short while. This was in 1935, since Schauder spoke about his journey to Moscow for an International Conference in Topology, in September 1935; the next year he went to Oslo for the International Congress of Mathematicians. His longest and most important trip abroad was, however, in 1932, before he came to our school. He had obtained a Rockefeller Foundation scholarship and went first to Leipzig (for the academic year 1932-1933) to Leon Lichtenstein, and later on (in April 1933) to Paris, where he collaborated with Jean Leray. The result was the famous memoir [7] for which Schauder and Leray obtained the grand international Metaxas prize for mathematics in 1938. Two other famous contributions were: The "Schauder fixed point principle" [8] (which he proved before he came to our school), and the "Schauder method in the boundary problems for partial differential equations" (this was done while he was a teacher in our school). The latter method was first developed in the classical setting of Holder spaces; it involved the derivation of the so-called Schauder estimates for linear elliptic second-order operators (cf. [10]). The same general ideas were used later by Schauder in another important direction: for proving the existence of generalized solutions to some nonlinear problems. This involved the introduction of so-called Sobolev spaces, the derivation of the Schauder estimates in this setting, and finally the applications of topological techniques (cf. [11], where the case of quasi-linear elliptic equations with continuous coefficients in R2 is discussed and [12], where quasi-linear hyperbolic equations are treated).

PHOTO 2. Juliusz Schauder in a group of participants of the First International Topology Conference in Moscow, September 4-10, 1935. (Top row: E. Cech, H. Whitney, K. Zarankiewicz, A. Tucker. S. Lefschetz, H. Freudenthal, F. Frankl, J. Nielsen, K. Borsuk. Bottom row: K. Kuratowski, J. Schauder, St. Cohn-Vossen, P. Heegaard, J. Rozanska).

Steinhaus comments on the development of Lwów mathematics in those years as follows (cf. [2], pp. 116-117):

*The character of Lwów mathematics changed conspicuously. Specialists
began to develop in different fields: Mazur worked in linear methods of
summability, Auerbach in differential geometry, Schauder in partial differential
equations, Kaczmarz and Orlicz in orthogonal series, Nikliborc in celestial
mechanics and figures of equilibrium; to mention only a few examples. *(
... )* It was possible to distinguish "schools". The school
of Banach was occupied with the theory of operators, the school of Kuratowski
*(*at Polytechnic up to 1934*)* with topology, the school of
Chwistek with mathematical logic; I tried to direct my students into the
direction of probability theory and applied mathematics. *

Steinhaus further mentions, pp. 157-158, that:

*In 1938 when Henri Lebesgue became doctor honoris causa at Lwów
University, in his official speech he mentioned that the Metaxas prize
was shared by Leray and Schauder. It was a great distinction for our docent
since the prize went to those who in recent years had made the most outstanding
contribution in mathematical analysis. Leray also visited us the same year;
he was a charming Frenchman of unusual elegance and culture.*

In January 1937, in the middle of the 7th grade, my father and I decided that I should quit our "classical" school and go over to a mathematics and natural science ("matematyczno-przyrodnicze") gymnasium. (It was the No. I State Gymnasium, named after Nicolas Copernicus. The No. III Gymnasium was at Batory Street 5 and the No. I, at Kubala Street 1. These streets formed a corner, and the buildings of both schools touched each other. The opposite corner was made with the short Fredro Street, at which other end the Scottish and Roma coffeehouses were located). Thus, for almost two years, I lost contact with Schauder.

It was October 1938 when we met again, in completely
different circumstances. In May 1938, I finished the Ist Gymnasium and
in October became a student of physics at the Jan Kazimierz University.
During the first year of study I had only two mathematics courses: one,
Professor Zylinski's *Introduction to Analysis* (with exercises),
and one with Professor Banach Analytical Geometry with Exercises (this
was, unfortunately, the only course I ever had from Banach).

*** * * **

The second academic year began after the outbreak of the Second World War: Poland was invaded in September 1939, first by the Germans, and then by the Soviets, and Lwów appeared to be in the Soviet zone. Lwów was included by force into the Ukrainian Soviet Republic and the University became the "Lviv State Ivan Franko University". Formally the Ukrainian language was introduced, but this was not strictly observed everywhere, at least during the first year. The first trimester started more or less according to previous Polish rules; later on the system was changed from trimestrial to semestrial, and all the studies were reorganized according to the Soviet model. In this first trimester I had the following lectures and exercises in mathematics:

Prof. H. Steinhaus | 4 hours | Ordinary Differential Equations |

Prof. H. Steinhaus | 2 hours | Seminar in Analysis |

Doc. J. Schauder | 4 hours | Theoretical Mechanics and Exercises |

Doc. H. Auerbach | 2 hours | Potential Theory and Exercises |

Doc. S. Mazur | 2 hours | Differential Geometry and Exercises |

I have Schauder's signature in my index book; it is the only material
thing I have directly from him. Unfortunately, I do not remember much of
his course, except that it was clear and systematic, in the rigorous mathematical
style of Leon Lichtenstein. Steinhaus recalls an interesting detail about
the Soviet occupation of Lwów (cf. [2], p. 182):
"*When Saks, Schauder and myself wrote down "Polish nationality"
in a form which everyone had to fill, it was said that we did this in fear
of the Polish police. But it was not said aloud, only whispered around*".
Although all three were of Jewish origin, they had proved their Polish
convictions also by their military service and by fighting for the Polish
cause: Steinhaus and Schauder as volunteers in World War I (Steinhaus in
Pilsudski's legions, Schauder in Haller's Blue Army), while Saks fought
in the Silesian Uprising in 1919 and also in September 1939 (I remember
him in an army coat).

In the winter and spring of 1940, deep changes occured at the university. My father was shifted from philosophy to German studies and the philosopher Ajdukiewicz to physics for medical students; but the teaching of mathematics and physics was not only increased by new Ukrainian and Russian professors and assistants, but also by some Polish refugees from Warsaw and Krakow. In physics the new professors were V. Milianchuk and Z. V. Chraplywy, both students of Rubinowicz. Steinhaus (cf. [2], p. 186) writes about this time:

*Banach became the dean of the Physico - Mathematical Faculty. He
recognized collaboration as the best method; maybe because of this it was
possible to arrange professorships for Saks and Knaster, which was not
easy since Soviets had a disinclination for people from Warsaw. There were
the following heads of chairs: Banach, Schauder, Zylinski, Zarzycki, Mazur
and myself; professors without chairs *(*the same as extraordinary
professors or associate professors*)*: Saks, Knaster, Chwistek, Auerbach
and Orlicz; docents: Eidelheit, Szpilrajn and Wojdyslawski. It was really
a very serious team and in normal times we could have done much with such
a group. It was possible to replace the *(*Polish*)* Mathematical
Society by introducing scientific sessions of the Faculty, and it was also
possible to publish volume IX of "Studia Mathematica" with the
only modifications that, except for the original text, each paper had to
have a Ukrainian summary. After Lawrentev's visit, some of us became collaborators
of the Kiev Academy* (*Banach, Schauder, Mazur, Saks, Zarzycki and
myself*)*. It was important for us since we became entitled to several
hundred rubles each month, which together with the university salary enabled
us to have a tolerably good life. *

Although the situation in physics and mathematical sciences looked well
and Lwów was one of the four best supplied cities in the Soviet
Union (with Moscow, Leningrad and Kiev), the general situation was quite
depressing. Massive deportation and arrests produced general uncertainty
and anxiety. Steinhaus writes ([2], p. 190): "*Everyday
I awoke with horror, when returning consciousness after the sleep reminded
me that I was in a prison." This explains his words above: "in
normal times we would ... *". Now the curriculum was strictly prescribed,
and I could not choose my courses freely as before. Schauder had the chair
of mechanics in the Department of Mathematics, but lectures of Theoretical
Mechanics for physicists were given by physicist Milianchuk. Schauder's
name did not occur in my new index-book; in mathematics, I only recognize
there now the following names: Auerbach, Stark, Raabe, Szpilrajn, Eidelheit,
Orlicz and Nikliborc. There was one exception in the rigid program of study:
a few professors led special advanced seminars with invited participants.
As I said at the beginning, I was invited by Schauder to his special seminar
on continuous groups. I remember the moment of this invitation well. It
was probably April or May of 1940 and I was just finishing my second year
of studies. I was sitting in the lecture room of the Department of Mathematics
and was preparing my exercises for mathematical analysis led by Dr. Antoni
Raabe, who was killed in 1942 by the Nazis in Auschwitz. Suddenly Professor
Schauder came in and asked me whether I would be interested in participating
in his seminar. I was surprised, since we had not had much contact in the
last few years. He was so friendly and full of smiles that I could not
refuse, although I did not know much about the continuous or Lie groups.
He said that we would start from the beginning, on the base of the new
Pontryagin book [15]. He also said that he would like
to have some theoretical physicist collaborate with him on these problems
in the future. I agreed to his proposal with pleasure, considering it a
distinction. In the next week Schauder secured a few copies of Pontryagin's
book, and gave me one (I have it up to now, it is a black book of yellowish
low quality paper, much thinner than the second Russian edition of 1954,
and it corresponds to the English translation in the American edition).

I do not remember when the seminar began; probably it started in the fall of 1940. It met two hours a week, and the subject was rather hard for me. There were During 1940 - 1941 the following Soviet mathematicians visited Lwów: P. S. Aleksandroff, M. Krein, L. Lusternik and S. Sobolev. On the other hand in 1940, Banach, Mazur, Schauder and Saks attended a conference on Functional Analysis in Kiev. only a few participants, and it seemed that for all, except Schauder, the topic was new. Pontryagin's book is wonderfully written, and with great pedagogical skill; nevertheless it was highly abstract for me at that time. It represented, however, my first contact with modern mathematics; a wonderful introduction into mathematical thinking and methods. I think that without this early schooling I would not have become a mathematical physicist. I therefore consider Schauder as one of my principal teachers; (the others were Rubinowicz in Lwów, and later on Weysenhoff and Golab in Krakow in 1945). The year 1940-1941 was the period of our closest contact. We met not only at the seminar, but frequently also on Zielona Street on our way home (I lived then at 19 Pawlikowski St., at the end of Zielona Street, and Schauder at 29 Tarnowski St., one of the side streets of Lwów. Our discussions (which were not merely casual) drew us closer to each other, and that in spite of the 21 years difference in age: he was then 41 and I was 20. But he did not tell me yet about his scientific plans in physics.

*** * ***

On June 22, 1941, the Germans attacked the Soviet Union and after 8 days Lwów was occupied by the German army. The University was closed, as well as most of the other schools. Immediately after occupying Lwów a special detachment of the German police troops killed a large group of Polish university professors and their families: 42 persons altogether, and among them 25 professors and docents (among mathematicians, the professors Ruziewicz, Bartel, Stozek and Lomnicki were killed). In October 1941, I started to work in a small optical factory on Zadworzanska Street, while my father became a teacher in a lower school of chemistry. I did not hear anything of Schauder. He disappeared somewhere, as did all the Jews and Poles of Jewish origin, because of their terrible persecution and the extermination by the Nazis. The persecution started immediately after the occupation of Lwów, but the establishment of the Jewish ghetto and the concentration and extermination camps (such as the Janowski camp in Lwów) took some time. So in the meantime some small percentage of Jews succeeded in hiding among the non-Jewish population which frequently helped them in spite of the great danger of being executed if discovered. I remember that in one of the first days of the German occupation I noticed a large detachment of German infantry and artillery in gigantic caterpillar transporters on Zielona Street. Suddenly I met there Doc. Max Eidelheit, one of my teachers of mathematics. He said to me that he, being of Jewish extraction, was hiding, and that Professor Schauder had probably also escaped any direct persecution so far; but he was not sure about Auerbach and others. Later on, all three were killed (Eidelheit in 1941 and Auerbach in 1942; about Schauder I will write below), as well as many other Polish mathematicians of Jewish and non-Jewish extraction: Kolodziejczyk, A. Lomnicki, Bartel, Stozek, Ruziewicz, Lindenbaum, Kerner, Saks, Schreier, Wojdyslawski.

Many months later-it was already 1943 - I came across Schauder himself in the street. I learned from A. Granas, that Professor S. Mazur, in his unpublished reminisccnces of Schauder, wrote that Schauder had visited him almost every week during 1943, and that they had had mathematical discussions (among others, about developing the results and methods in the variational calculus intiated in their joint Note during the Oslo Congress); one of Schauder's other mathematical interests at that time was quantum mechanics. On the other hand, according to K. Maurin, Professor W. Rubinowicz once told him that during this time of hiding from the Germans in Lwów, Schauder visited him several times (on the corner of Zielona and Pawlikowski Streets) and asked him some mathematical questions about the initial value problem for differential equations, in which Rubinowicz was also an expert. As a student of the Clandestine University, I, as well as my father, was also a frequent visitor at the Rubinowicz home at that time; but I never met Schauder there, and I do not remember if Rubinowicz told me about his visit. The comments of these professors show that, in spite of the prevailing terrible conditions, Schauder was working - or at least trying to work - in mathematics. He looked pale and yellowish, but not much worse than usual. We were on lower Grodecka Street; we walked along slowly, observing in the distance the green hill of St. Yur (Jur=George) with its Greek-Catholic cathedral; and below it was a crowd swarming around something (a market place, or a circus?), so probably it was early autumn. I was made very uneasy because of his rather semitic appearance (as it seemed to me at that moment), and I told him that it was dangerous for him to go about on the streets, where many people, and especially the police, could recognize him as a Jew. He said that he looked "Aryan", but that although some danger existed, of course, he could not stay at home all the time, where he had to lie hidden in a covered bed during the daytime. He said that his wife and daughter were in Warsaw and that he wanted to go there by any means. (His wife was killed by the Nazis; his daughter escaped this fate, being rescued by nuns in a cloister). I said that Warsaw was not less dangerous than Lwów and that the journey would be dangerous too. But he asked me if I could help him to get false documents of high quality, since he had no good papers. I promised to do what I could, and we parted.

At that time, as well as before the war, I was a member of the ZNMS (the Society of Independent Socialist Youth, which was a subsidiary of the Polish Socialist Party, the PPS). The PPS, which was then underground, was one of the four political parties that formed the Polish Exile Government in London, and the underground Delegate of the Government for Lwów was also a member of the PPS (Dr. Adam Ostrowski, later Director of the Publishing House PIW, now deceased). With the help of my colleagues from the organization (Przemyslaw Ogrodzinski, Maryna Fiderer and others) it was easy to organize help for Schauder. They contacted him and he obtained excellent documents, probably also some money, tickets and other help. (Active in this assistance was the Department for Helping Jews "Zegota", of the AK, the Home Army to which we all belonged). A few days before Schauder's planned departure for Warsaw, we met again. He thanked me for everything and said that the only thing he still needed was the possibility to take a bath before his journey. In the home where he was hiding, there was no bathroom. I said that in my apartment (I lived then in a small room at 10 Rycerska Street) there was no bathroom either, but that in the factory where I was employed there was a conveniently located one, so he could come and have a bath. But on the fixed day and hour he did not arrive. Shortly afterwards, I learned through our organization what had happened. Schauder was on his way to Zadworzanska Street when he was suddenly arrested. He was taken to a prison with which we had some clandestine contact, and through the organization some messages and help were transmitted to him. But a few days later a transport of Jews was taken from prison to the Kleparow Railway Station. There Schauder tried to escape from the police and was killed on the spot. This was probably in October 1943.

In the present year 1993, exactly 50 years have elapsed since Juliusz Schauder's premature and cruel death. 50 years is longer than his life-time - he lived only 44 years. During these 50 years his name has become known among mathematicians and mathematical physicists as one of the great names in the history of mathematics. He animated mathematics through unifying work in four branches of mathematics, not through superficial but by deep threads. We know that he wanted to go further in the direction of theoretical physics, but his death prevented the full realization of his life dream. The memory of this great man and mathematician will be cherished forever.

**REFERENCES**

ROMAN S. INGARDEN

Institute of Physics

Nicholas Copernicus University

Torun, POLAND