The Birth of Functional Analysis

Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. Together they founded what would, in 1920, become the Polish Mathematical Society.

A coin and a postage stamp commemorating Stefan Banach.

A coin and a postage stamp commemorating Stefan Banach.

Banach returned to Lvov in 1920 and earned a doctorate for a very original thesis which marked the birth of a new branch of mathematics called functional analysis. We may think of a vector as an arrow in the plane or in three-dimensional space. Vectors are added by placing them head-to-tail; they form a structure called a linear space. Since each vector has a definite length, called its norm, we speak of a normed linear space.

Banach generalized this idea to an infinite number of dimensions and introduced the additional technical condition of completeness: In a Cauchy sequence, the terms become arbitrarily close to each other as the sequence progresses. For a complete space, every Cauchy sequence has a limit. Essentially, completeness ensures there are no “gaps” in the space. The complete normed linear function spaces that Banach defined now bear his name.

Banach spaces, or complete normed linear spaces, provide an ideal setting for a wide range of problems in both pure and applied mathematics. Banach applied his ideas to an area of mathematics called integral equations. He was not interested in applications outside mathematics, but functional analysis proved to be an immediate and spectacular success as a foundation for the emerging physical theory of quantum mechanics.

Stefan Banach (1892–1945)

Stefan Banach (1892–1945)

Together with Steinhaus, Banach started a new journal, Studia Mathematica, with emphasis on the new field of functional analysis. Through Steinhaus, Banach met Lucja Braus, who was to become his wife. They married in 1920 and Lucja was his faithful companion for the remaining twenty five years of his life.

The mathematicians in Lvov did a lot of their work in coffee houses, most notably in The Scottish Café. The problems they raised were written in a book kept by the proprietor of the café. A remarkable range of new theorems, and many challenging problems, were listed in “The Scottish Book”.

Banach was allowed to continue his research and lecturing after the occupation of Lvov by the Soviet Union in 1939, and he maintained contacts with mathematicians in Russia. But life became extremely difficult after the Nazi invasion in 1941. Banach managed to survive the war but he died of lung cancer shortly afterwards in 1945.

Banach proved several important results in functional analysis and a number of theorems bear his name. One of the most amazing is the Banach-Tarski theorem, which states that a spherical ball can be divided into a small number of pieces or subsets which fit together to from two balls of radius equal to the first sphere. It is among the most paradoxical results in all of mathematics.

Banach’s achievements earned him international recognition and he won several prestigious prizes and honours. He was one of the “greats” of twentieth century mathematics and is still considered a hero in Poland.

[ Next week: More about the Lvov Schoool of Mathematics ]

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