Banach came from a poor family, and he had very little conventional schooling at first. He was largely self-taught when he arrived at the Polytechnical Institute. It is said that Steinhaus accidentally discovered his talent when he overheard a mathematical conversation between two young students sitting on a park bench. One was Banach, the other Nikodym, now recently retired as professor of mathematics at Kenyon College. Banach and Steinhaus were to become the closest of collaborators and the founders of the Lwów school of mathematics.
Banach's knowledge of mathematics was broad. His contributions were in the theory of functions of real variables, set theory and, above all, functional analysis, the theory of spaces of infinitely many dimensions (the points of these spaces being functions or infinite series of numbers). They include some of the most elegant results. He once told me that as a young man he knew the three volumes of Darboux's Differential Geometry.
I attended only a few of Banach's lectures. I especially remember some on the calculus of variations. In general, his lectures were not too well prepared; he would occasionally make mistakes or omissions. It was most stimulating to watch him work at the blackboard as he struggled and invariably managed to pull through. I have always found such a lecture more stimulating than the entirely polished ones where my attention would lapse completely and would revive only when I sensed that the lecturer was in difficulty.
Beginning with the third year of' studies, most of my mathematical work was really started in conversations with Mazur and Banach. And according to Banach some of my own contributions were characterized by a certain "strangeness" in the formulation of' problems and in the outline of possible proofs. As he told me once some years later, he was surprised how often these "strange" approaches really worked. Such a statement, coming from the great master to a young man of twenty-eight, was perhaps the greatest compliment I have received.
In mathematical discussions, or in short remarks he made on general subjects, one could feel almost at once the great power of his mind. He worked in periods of great intensity separated by stretches of apparent inactivity. During the latter his mind kept working on selecting the statements, the sort of alchemist's probe stones that would best serve as focal theorems in the next field of study.
He enjoyed long mathematical discussions with friends and students. I recall a session with Mazur and Banach at the Scottish Café which lasted seventeen hours without interruption except for meals. What impressed me most was the way he could discuss mathematics, reason about mathematics, and find proofs in these conversations.
Since many of these discussions took place in neighborhood coffee houses or little inns, some mathematicians also dined there frequently. It seems to me now the food must have been mediocre, but the drinks were plentiful. The tables had white marble tops on which one could write with a pencil, and, more important, from which notes could be easily erased.
There would be brief spurts of conversation, a few lines would be written on the table, occasional laughter would come from some of the participants, followed by long periods of silence during which we just drank coffee and stared vacantly at each other. The café clients at neighboring tables must have been puzzled by these strange doings. It is such persistence and habit of concentration which somehow becomes the most important prerequisite for doing genuinely creative mathematical work.
Thinking very hard about the same problem for several hours can produce a severe fatigue, close to a breakdown. I never really experienced a breakdown, but have felt "strange inside" two or three times during my life. Once I was thinking hard about some mathematical constructions, one after the other, and at the same time trying to keep them all simultaneously in my mind in a very conscious effort. The concentration and mental effort put an added strain on my nerves. Suddenly things started going round and round, and I had to stop.
These long sessions in the cafés with Banach, or more often with Banach and Mazur, were probably unique. Collaboration was on a scale and with an intensity I have never seen surpassed, equaled or approximated anywhere — except perhaps at Los Alamos during the war years.
Banach confided to me once that ever since his youth he had been especially interested in finding proofs — that is, demonstrations of conjectures. He had a subconscious system for finding hidden paths — the hallmark of his special genius.
After a year or two Banach transferred our daily sessions from the Café Roma to the "Szkocka" (Scottish Café) just across the street. Stozek was there every day for a couple of hours, playing chess with Nikliborc and drinking coffee. Other mathematicians surrounded them and kibitzed.
Kuratowski and Steinhaus appeared occasionally. They usually frequented a more genteel teashop that boasted the best pastry in Poland.
It was difficult to outlast or outdrink Banach during these sessions. We discussed problems proposed right there, often with no solution evident even after several hours of thinking. The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed in the meantime. If they were not polished or even not quite correct, Mazur would frequently put them in a more satisfactory form.
Needless to say such mathematical discussions were interspersed with a great deal of talk about science in general (especially physics and astronomy), university gossip, politics, the state of affairs in Poland; or, to use one of John von Neumann's favorite expressions, the "rest of the universe." The shadow of coming events, of Hitler's rise in Germany, and the premonition of a world war loomed ominously.
Banach's humor was ironical and sometimes tinged with pessimism. For a time he was dean of the Faculty of Science and had to attend various committee meetings. He tried to avoid all such activities, as much as he could, and once he told me, "Wiem gdzie nie bede [I know where I won't be]," his way of saying that he did not intend to attend a dull meeting.
Banach's faculty for proposing problems illuminating whole sections of mathematical disciplines was very great, and his publications reflect only a part of his mathematical powers. The diversity of his mathematical interests surpassed that shown in his published work. His personal influence on other mathematicians in Lwów and in Poland was very strong. He stands out as one of the main figures of this remarkable period between the wars when so much mathematical work was accomplished.
I have had no precise knowledge of his life and work from the outbreak of the war to his premature death in the fall of 1945. From fragments of information obtained later, we learned that he was still in Lwów during the German occupation and in miserable circumstances. Surviving to see the defeat of Germany, he died in 1945 of lung disease, probably cancer. I had often seen him smoke four or five packs of cigarettes in a day.
In 1929 Kuratowski asked me to participate in a Congress of Mathematicians from the Slavic Countries which was to take place in Warsaw. What sticks in my mind is a reception in the Palace of the Presidium of the Council of Ministers and my timidity at seeing so many great mathematicians, government officials, and important people. This was overcome somewhat when another mathematician, Aronszajn, who was four or five years older than I, said, "Kolego" (this was the way Polish mathematicians addressed each other), "let's go to the other room, the pastry is very good there." (He is now a professor at the University of Kansas in Lawrence.)