Banach
# Stefan Banach

### Born: 30 March 1892 in Kraków, Austria-Hungary (now Poland)

Died: 31 Aug 1945 in Lvov, Ukraine

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**Banach** went to school in Kraków. On leaving school he wanted to work in a subject other than mathematics but soon changed his mind. He lectured in mathematics at the Institute of Technology in Lvov (1919) and at the University of Lvov (1922) becoming a professor at the University of Lvov (1927).

Banach spent the Second World War living under difficult conditions under the Nazi occupation in Lvov. He planned to go to Kraków after the war to take up the chair of mathematics at the Jagiellonian University but he died in 1945 of lung cancer.

Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces. In addition, he contributed to measure theory, integration, and orthogonal series.

In his dissertation, written in Lvov in 1920, he defined axiomatically what today is called a Banach space. The name 'Banach space' was coined by Fréchet. Banach algebras were also named after him.

A Banach space is a real or complex normed vector space that is complete as a metric space under the metric

d(x,y) = ||x-y||

induced by the norm. The completeness is important as this means that Cauchy sequences in Banach spaces converge.
A Banach algebra is a Banach space where the norm satisfies

||xy|| ||x||.||y||

Many important theorems are named after Banach. There is the Hahn-Banach theorem, the Banach-Steinhaus theorem, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball. His open mapping of 1929 used set-theoretic concepts introduced by Baire in his 1899 dissertation.
Banach's most important work is the *Théorie des opérations linéaires* (1932).

**References (18 books/articles)**

**References elsewhere in this archive:**

Tell me about Banach's work on abstract linear spaces

Tell me about Banach's work on topology

JOC/EFR December 1996