Stefan Banach | Biography

In spite of his somewhat fragmented education (he never completed a formal doctoral program), Stefan Banach made important contributions to a number of fields of mathematics, including the theory of orthogonal series, topology, the theory of measure and integration, set theory, and the theory of linear spaces of an infinite number of dimensions. He is probably best remembered, however, for his work on functional analysis.

Stefan Banach was born on March 30, 1892, in Kraków, Poland. His father was named Greczek, a railway official from peasant background. He and Stefan's mother (whose name has been lost) abandoned their young child to a laundress almost immediately after his birth. The child took on his foster mother's surname of Banach, but almost nothing else is known about his early childhood. Banach apparently developed an interest in mathematics at an early age and taught himself the fundamentals of the subject. By the age of 15 he was supporting himself as a private teacher of mathematics. He also taught himself enough French to master Tannery's text on the theory of functions, Introduction à la théorie des fonctions. Banach attended lectures on mathematics at Jagellon University on an irregular basis before entering the Lwów Institute of Technology in the Ukraine in 1910. He did not, however, graduate from the institute.

In 1914, with the outbreak of World War I, Banach returned to Kraków. Two years later, a chance event was to change his life. While sitting on a park bench in Kraków talking with a friend about mathematics, he was overhead by the mathematician H. Steinhaus. Steinhaus later wrote that he was "so struck by the words 'the Lebesgue integral'" that he heard from the two that he came closer and introduced himself to the young men. As the group talked, the conversation turned to a problem on the congruence of a Fourier series on which Steinhaus had been working. "I was greatly surprised," Steinhaus went on to say, "when, after a few days, Banach brought me a negative answer with a reservation which resulted from his ignorance of [a technical point about which he did not know]." Banach and Steinhaus were later to collaborate on a number of mathematical studies.

Banach's natural gift for mathematics soon became more widely known, and at the conclusion of the war he was offered a position as mathematical assistant at the Lwów Institute of Technology by Antoni Lomnicki. For the first time in his life Banach had some degree of financial security and he married. Beginning in 1919, Banach was assigned to lecture on mathematics and mechanics. In the same year, he was awarded his doctoral degree although he had not completed the full program of courses expected for that degree.

Publishes Historic Paper on Integral Equations

The primary basis for Banach's degree was the paper he had written on integral equations, which had been published in Fundamenta mathematicae in 1922 as "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales." At about the same time he was made an instructor at the institute; in 1927 he was promoted to full professor. From 1939 to 1941 Banach also served as dean of the faculty at the institute.

The mathematical work for which Banach is best known is his book Théorie des opérations linéaires, which appeared in 1932 as the first volume in the Mathematical Monographs series, published in Warsaw. In this book, Banach developed a general theory for working with linear operations that proved to be a landmark in the field. Prior to this work, a number of individual, discrete methods had been developed for solving specific problems. But there was no comprehensive theory that could be applied to a great variety of problems. In his book, Banach introduced the concept of normed linear spaces, now known as Banach spaces, which, Steinhaus later wrote, can be used "to solve in a general way many problems which formerly called for special treatment and considerable ingenuity."

Banach's significance in the history of mathematics goes beyond his own research. He was also an effective teacher whose influence was spread throughout Europe and the United States by a number of brilliant students. In addition, he wrote an important popular textbook, Differential and Integral Calculus (1929-30) and was founder with Steinhaus of the journal Studia mathematica.

World War II was a personal disaster for Banach. After the German army occupied the city of Lwów, he was forced to work in a German laboratory studying infectious diseases. His job there was to feed the lice used in experiments. As degrading as this work was, Banach was able to continue teaching in underground schools and carry on his own research. By the time the war ended, however, his health had so badly deteriorated that he lived only a few more months. Banach died in Lwów on August 31, 1945. Among the honors accorded him during his lifetime were election as corresponding member of the Polish Academy of Sciences and of the Kiev Academy of Sciences. He also received the Prize of the City of Lwów in 1930 and the Prize of the Polish Academy in 1939. Upon his death, the city of Warsaw renamed one of its streets in his honor.