The Life of Stefan Banach

Sheldon Axler

This review of Roman Kaluza’s 1996 book The Life of Stefan Banach was published in American Mathematical Monthly 104 (1997), 577-579.

In at least one printing of the current (fifteenth) edition of the Encyclopedia Britannica, the entry on Stefan Banach did not contain the words "Poland" or "Polish". The Britannica called Banach a "Soviet mathematician". The encyclopedia fixed its error in later printings, but the mathematics community has not yet adequately documented Banach’s life and ideas. A computer search of Mathematical Reviews reveals more than eleven thousand publications with the word "Banach" in the title; "Hilbert" occurs in only seven thousand titles. Yet no mathematician or historian of mathematics has produced a book-length biography of Stefan Banach.

The book under review was written neither by a mathematician nor by a historian. The author, a Polish reporter and journalist, writes well about mathematics without using any mathematical symbols. Professional mathematicians will spot a few technical errors of the type that inevitably creep into exposition at this level. For example, we read that "the only linear transformations" on a finite-dimensional Euclidean space are "translations, rotations, and reflections". Such small mistakes in mathematical details can easily be forgiven because the author does a good job of capturing the flavor of early functional analysis and its creators.

The book suffers more from the lack of a historian’s perspective than from an absence of mathematical expertise. Some events described in the book cry out for more explanation. For example, consider the author’s description of the Nazi efforts to eliminate the intelligentsia in occupied Poland during World War II. Before capturing the Polish university town of Lvov, where Banach lived and worked, German officials compiled a list of prominent professors, scientists, and writers in Lvov who would be executed. One night shortly after German soldiers had entered Lvov, SS units murdered forty leading intellectual figures in Lvov without even the pretense of trials. But Banach was untouched by the Nazi death squads. An alert reader will wonder why Banach, who at this time was President of the Polish Mathematical Society and a Dean at the university, was not among the intellectuals marked down for liquidation. Unfortunately the author does not comment on the apparent disparity between his description of Nazi plans to crush Polish intellectual life and the survival of Banach, Poland’s most influential mathematician. Was Banach spared because he had too much fame? Or were the occupying forces so mathematically illiterate that they had never heard of Banach? The author does not even speculate about these questions that beg to be answered.

As another example of a tantalizing tidbit from the book that needs more explanation, consider the following account (page 51) of Banach’s support for the mathematical logician Leon Chwistek:

... when at some point Chwistek applied for a position in logic in Lvov, Banach backed him unequivocally and helped him to obtain the post. The affair scandalized half of intellectual Poland since Chwistek, in addition to being a respected scholar, also had a well-deserved reputation as being a somewhat strange and very eccentric artist.

Banach himself was "somewhat strange" and "eccentric"; that description surely fits many mathematicians. So why would Banach’s support for such a person have "scandalized half of intellectual Poland"? Readers will realize that something more must have been involved here, but the author provides no hints to help solve this mystery.

In 1928 Stefan Banach and his colleague Hugo Steinhaus founded Studia Mathematica, which quickly became the most important journal specializing in the then new field of functional analysis. Today’s mathematics librarians, grappling with budget problems, will be amused to learn that the first volume of Studia Mathematica cost $1.50 outside Poland.

When teaching the graduate course in functional analysis, I always use the Krein-Milman Theorem and its appearance in Studia Mathematica as an excuse to inject a bit of history into the classroom. The Krein-Milman Theorem states that in a locally convex topological vector space, every compact convex set is the closed convex hull of its extreme points. This result was published (in somewhat less generality than the version just stated) in the 1940 volume of Studia Mathematica, which also contained two papers written by Banach. That volume of the journal was printed on poor-quality paper, clearly due to wartime conditions. The most curious feature of the 1940 volume is that each article (they are all written in either English, French, or German) appears with an abstract in Russian. Obviously Lvov, where Studia Mathematica was published, lay in the Soviet zone of occupation at the time of publication. Two weeks after Germany had invaded Poland from the west in September 1939, the Soviet Union marched into Poland from the east. Poland was partitioned between Germany and the Soviet Union until the summer of 1941, when Germany attacked the Soviet Union and occupied all of Poland.

The 1940 volume of Studia Mathematica was the last one edited by Banach, who died at age 53 shortly after World War II ended in 1945. After an absence of eight years, Studia Mathematica resumed publication in 1948 in Wroclaw. Poland’s border had moved westward after World War II, so that Lvov was then in the Soviet Union (no doubt this accounts for the Britannica’s claim that Banach was a "Soviet mathematician"). A few years ago Lvov again changed countries---it is now part of Ukraine. Today Studia Mathematica, still a fine journal specializing in functional analysis, is published in Warsaw. The cover of each issue still proudly bears the names of the founding editors Banach and Steinhaus.

In 1932 Banach published his famous book Théorie des Opérations Linéaires, based on his Polish version published a year earlier. Remarkably, Théorie des Opérations Linéaires remains in print today more than six decades after its original publication, partly because of its historic value as the first monograph on functional analysis but also because of the clean, modern style with which Banach presents the fundamentals of the subject (as created in good part by him and his collaborators). While a graduate student, I read Théorie des Opérations Linéaires to study for my French exam. I remember the thrill of seeing functional analysis developed by a legendary hero of twentieth century mathematics and my delight in his extraordinarily clear writing. I also remember my amusement that what we today call "Banach spaces" are called "spaces of type (B)" in Banach’s book. From the book under review I learned that Banach had previously written several popular high school mathematics textbooks for use throughout Poland; perhaps writing for a high school audience had honed Banach’s excellent expository skills.

The Life of Stefan Banach left me hungry for more information about this fascinating figure. However, the author has performed a valuable service by uncovering some previously unknown data about Banach and by interviewing many of the dwindling number of people who knew Banach. This sketchy biography is a good place to start for someone wanting to learn about Banach.