VI.84 Stefan Banach

b. Kraków, Poland, 1892; d. Lwów, Poland, 1945

Functional analysis; real analysis; measure theory;
orthogonal series; set theory; topology

Banach was the son of Katarzyna Banach and Stefan Greczek. As his parents were unmarried and his mother was too poor to support her son, he was brought up mainly in Kraków by a foster mother, Franciszka Płowa.

After graduating from high school in 1910, Banach enrolled at the Lwów Polytechnic in the Faculty of Engineering. Two years after his studies were interrupted by the outbreak of World War I, Banach returned to Kraków, where on a summer evening in 1916 he was “discovered” by Hugo Steinhaus, who overheard the words “Lebesgue integral” and brought him to Lwów. Steinhaus considered this event as his “greatest mathematical discovery.” It was also through Steinhaus that Banach met his future wife, Łucja Braus, whom he married in 1920.

In the same year Professor Antoni Łomnicki engaged Banach as his assistant at the Lwów Polytechnic, even though Banach had not yet finished his studies. This was the beginning of the meteoric rise of Banach’s scientific career.

In June 1920 Banach defended his doctoral dissertation, On operations on abstract sets and their application to integral equations,” at the Jan Kazimierz University in Lwów. His dissertation was written in Polish and published in 1922 in French. In his thesis Banach introduced the concept of complete normed linear spaces, which are today known as BANACH SPACES [III.62] (the name was proposed by Fréchet in 1928). The theory combined the contributions of RIESZ [VI.74], Volterra, FREDHOLM [V1.66], Lévy, and HILBERT [VI.63] On concrete spaces and on integral equations into a general theory. Banach’s dissertation could be viewed as the birth of functional analysis, since Banach spaces are one of its central objects of study.

On April 17, 1922, the Jan Kazimierz University in Lwów awarded Banach his habilitation (a degree allowing him to teach at the university), after which he was appointed Docent in Mathematics. On July 22, 1922, he became a professor of the university (and a full professor from 1927). Banach achieved great research results and became an authority in functional analysis and MEASURE THEORY [III.55]. During the academic year 1924–25 Banach was on sabbatical leave in Paris, where he met LEBESGUE [VI.72], who became a lifelong friend.

In Lwów a group of talented young mathematicians around Banach and Steinhaus soon became the Lwów School of Mathematics and started the journal Studia Mathematica in 1929. Among the members of this school were S. Mazur, S. Ulam, W. Orlicz, J. P. Schauder, H. Auerbach, M. Kac, S. Kaczmarz, S. Ruziewicz, and W. Nikliborc. Banach also collaborated with Steinhaus, Saks, and Kuratowski. Many of these mathematicians were later killed by the Germans during the occupation of Poland.

In 1932, Banach’s famous book Theory of Linear Operations appeared in French (a Polish version was published the year before) as part of a new series of mathematical monographs, of which he was one of the founders. This was the first monograph on functional analysis as an independent discipline, and it was the culmination of more than a decade of intense activity by Banach and others.

Banach and the mathematicians around him liked to discuss mathematics in the Café Szkocka (“Scottish café”). This unconventional way of doing mathematics made the atmosphere of Lwów unique—it is one of the rare cases in mathematics of genuine teamwork among a large group. Turowicz and Ulam noted that (see Kaluza 1996, pp. 62, 74):

Banach liked to spend most of his days in a café. He liked the noise and the music. They did not prevent him from concentrating and thinking. It was difficult to outlast or outdrink Banach during these sessions. Problems posed right there were discussed, 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.

One day in 1935, Banach proposed that the open problems should be collected in a notebook. This notebook later became famous under the name The Scottish Book.” In the years 1935–41 over 190 problems from various branches of mathematical analysis were proposed in this notebook, and the collection was published in English in 1957 by Ulam. A version with commentaries was published in 1981 by Birkhäuser as The Scottish Book, Mathematics from the Scottish Café (edited by R. D. Mauldin).

Banach was also the author of the books Mechanics (in two volumes, 1929 and 1930; English translation in 1951), Differential and Integral Calculus (in two volumes, 1929 and 1930, with several editions in Polish), Introduction to the Theory of Real Functions (in two volumes, written by Banach before the war, although only the first volume remains), and ten textbooks (jointly written with Stoek and SIERPSKI [VI.77]) for primary and secondary schools on arithmetic, geometry, and algebra (published in the years 1930–36 and reprinted in 1944–47).

Banach’s famous discoveries in functional analysis had three important steps. First, he considered abstract linear spaces, where functions are treated like points or vectors, sets of functions as function spaces, and operations on functions as operators. Second, he introduced the norm || · || of a mathematical object, that is, a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. The distance between two abstract elements x and y is then given naturally by d (x,y) = ||x – y ||. The third important step was to introduce the notion of “completeness” for these spaces. In such general spaces (Banach spaces) he was able to prove several fundamental theorems, like the uniform boundedness principle, the open mapping theorem, and the closed graph theorem. What these results say, roughly speaking, is that in a Banach space we cannot have bad (pathological) behavior everywhere—there is always some part of the space where our linear map or other object is well-behaved.

Names like Banach space, Banach algebra, Banach lattice, Banach manifold, Banach measure, the Hahn–Banach theorem, the Banach fixed point theorem, the Banach–Mazur game, the Banach–Mazur distance between isomorphic spaces, Banach limits, the Banach–Saks property, the Banach–Alaoglu theorem, and the BANACH–TARSKI PARADOX [V.3] show how wide his influence has been. Banach also introduced the notions of DUAL SPACE [III.19], dual operator, and the general concepts of weak and weak-star convergence, and he used all of these notions in linear operator equations.

In 1936, Banach delivered a one-hour plenary address at the International Congress of Mathematicians in Oslo, where he described the work of the whole Lwów school. In 1937 Norbert Wiener tried to lure him to the United States. In 1939 he was elected president of the Polish Mathematical Society and was awarded a Grand Prize of the Polish Academy of Knowledge. Banach spent the war years in Lwów. During the years 1940–41 and 1944–45 he was the Dean of the Faculty of Science at the renamed Iwan Franko State University. In the period 1941–44 Lwów was occupied by the German army. During this period Banach was saved from almost certain death by the action of Rudolf Weigel, a “Schindleresque” factory owner and inventor of the typhus vaccine, who gave him employment at his Bacteriological Institute as a louse feeder. After the war, he accepted a chair at the Jagiellonian University. He died on August 31, 1945, in Lwów of lung cancer at the age of fifty-three.

The complete list of Banach’s publications comprises fifty-eight items, and they were reprinted in Banach’s Collected Works (published in two volumes in 1967 and 1979). Banach said, “Mathematics is the most beautiful and most powerful creation of the human spirit. Mathematics is as old as Man.” Banach is considered a national hero in Poland, as a great scientist and a major figure in the great flowering of Polish scientific life in the independent Poland of the interwar years.

Further Reading

Banach, S. 1967, 1996. Oeuvres, two volumes. Warsaw: PWN.

Kaluza, R. 1996. The Life of Stefan Banach. Basel: Birkhäuser.

Lech Maligranda