**Hans Hahn** was an Austrian mathematician who is best remembered for the Hahn-Banach theorem. He also made important contributions to the calculus of variations, developing ideas of Weierstrass.

- Hans was a student at Strasbourg, Munich and Göttingen.
- Hahn was appointed to the teaching staff in Vienna as a privatdozent in 1905 after submitting his habilitation thesis.
- In session 1905-06 Hahn substituted for Otto Stolz at Innsbruck.
- As Menger explains, Hahn was a pioneer in set theory and functional analysis.
- However to many mathematicians he is best remembered for the Hahn-Banach theorem which we mention again below.
- In 1923 he introduced what today is known as the Hahn sequence space.
- Hahn wrote four papers on functional analysis.
- These include a report on integral equationS he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927.
- Other papers in this area characterise topological spaces that are continuous images of a line segment and related to this topic is what is now known as the Hahn-Mazurkiewicz theorem.
- Another area on which Hahn did research was measure theory.
- Fourier analysis also interested Hahn, and he looked at singular integrals and orthogonal expansions investigating the validity of the Parseval relation in various circumstances.
- Despite this wealth of deep papers on a wide range of mathematical topics, many people think of Hahn as a mathematical philosopher.
- During the 1920s Hahn, together with Philipp Frank, Otto Neurath, Moritz Schlick, founded the Vienna Circle of Logical Positivists, a discussion group of gifted scientists and philosophers who met regularly in Vienna.
- Menger writes on Hahn's role in the Vienna Circle in the article Hahn and the Vienna Circle.
- However, Hahn's ideas on mathematical intuition are strongly criticised by Benoit Mandelbrot in an article he wrote in 1982.
- My reaction is very different: Fractal geometry demonstrates that Hahn was dead wrong.
- Hahn received many honours for his achievements including the Lieban Prize in 1921.
- Although Hahn died in 1934, he still managed to be an author of the book Set Functions which was published in 1948, fourteen years after his death.

Born 27 September 1879, Vienna, Austria. Died 24 July 1934, Vienna, Austria.

View full biography at MacTutor

Origin Austria

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive