Hans Hahn (; 27 September 1879 – 24 July 1934) was an

Austria
Austria (, ; german: Österreich ), officially the Republic of Austria (german: Republik Österreich, links=no, ), is a landlocked
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n mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
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and philosopher
A philosopher is someone who practices philosophy
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who made contributions to functional analysis
200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.
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, topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
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, set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...

, the calculus of variations
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...

, real analysis
200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

, and order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...

. In philosophy he was among the main logical positivists
Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, was a movement in Western philosophy
Western philosophy refers to the philosophy, philosophical thought and work of the Western worl ...

of the Vienna Circle
The Vienna Circle (german: Wiener Kreis) of Logical Empiricism was a group of philosophers and scientists drawn from the Natural science, natural and Social Sciences, social sciences, logic and mathematics who met regularly from 1924 to 1936 at th ...

.
Biography

Born inVienna
Vienna ( ; german: Wien ; bar, Wean, label= ) is the , largest city, and one of of . Vienna is Austria's , with about 2 million inhabitants (2.6 million within the , nearly one third of the country's population), and its , , and centre. It ...

as the son of a higher government official of the k.k. Telegraphen-Korrespondenz-Bureau (since 1946 named "Austria Presse Agentur"), in 1898 Hahn became a student at the Universität Wien starting with a study of law. In 1899 he switched over to mathematics and spent some time at the universities of Strasbourg
Strasbourg (, , ; german: Straßburg ; gsw, label= , Strossburi , gsw, label= , Strossburig ) is the and largest city of the of and the of the . Located at the with Germany in the historic region of , it is the prefecture of the .
I ...

, Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of Bavaria
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and Göttingen
Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony
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. In 1902 he took his Ph.D. in Vienna, on the subject "Zur Theorie der zweiten Variation einfacher Integrale". He was a student of .
He was appointed to the teaching staff (Habilitation
Habilitation is the procedure to achieve the highest university degree in many European countries in which the candidate fulfills certain criteria set by the university which require excellence in research, teaching, and further education. Its qua ...

) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz at Innsbruck
Innsbruck (; bar, Innschbruck, label=Bavarian language, Austro-Bavarian ) is the capital of Tyrol (state), Tyrol and the List of cities and towns in Austria, fifth-largest city in Austria. On the Inn (river), River Inn, at its junction with the W ...

and some further years as a Privatdozent
''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualifica ...

in Vienna, he was nominated in 1909 Professor extraordinarius
Academic ranks in Germany are the titles, relative importance and power of professors, researchers, and administrative personnel held in academia.
Overview
Appointment grades
* (Pay grade: ''W3'' or ''W2'')
* (''W3'')
* (''W2'')
* (''W2'', onl ...

in Czernowitz
Chernivtsi ( uk, Чернівці́} ; ro, Cernăuți; see also ) is a city in . It is situated on the upper course of the , and is the of (), which includes the Ukrainian part of . Chernivtsi is also the administrative center of and host ...

, at that time a town within the empire of Austria. After joining the Austrian army in 1915, he was badly wounded in 1916 and became again Professor extraordinarius
Academic ranks in Germany are the titles, relative importance and power of professors, researchers, and administrative personnel held in academia.
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Appointment grades
* (Pay grade: ''W3'' or ''W2'')
* (''W3'')
* (''W2'')
* (''W2'', onl ...

, now in Bonn
The Federal city
The term federal city is a title for certain cities in Germany
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. In 1917 he was nominated a regular Professor
Professor (commonly abbreviated as Prof.) is an academic
An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary education, secondary or tertiary education, tertiary higher education, highe ...

there and in 1921 he returned to Vienna with this title, where he stayed until his rather early death in 1934 at the age of 54, following cancer surgery.
He had married Eleonore ("Lilly") Minor in 1909 and they had a daughter, Nora (born 1910).
He was also interested in philosophy, and was part of a discussion group concerning 's positivism
Positivism is a philosophical theory
A philosophical theory or philosophical position''Dictionary of Theories'', Jennifer Bothamley is a view that attempts to explain or account for a particular problem in philosophy
Philosophy (from ...

with Otto Neurath
Otto Karl Wilhelm Neurath (; 10 December 1882 – 22 December 1945) was an Austria
Austria (, ; german: Österreich ), officially the Republic of Austria (german: Republik Österreich, links=no, ), is a landlocked Eastern Alps, East Alp ...

(who had married Hahn’s sister Olga Hahn-Neurath in 1912), and Phillip FrankPhilipp Frank (March 20, 1884, Vienna, Austria-Hungary – July 21, 1966, Cambridge, Massachusetts, Cambridge, Massachusetts, United States) was a physicist, mathematician and also a philosopher during the first half of the 20th century. He was a log ...

prior to the First World War. In 1922, he helped arrange Moritz Schlick, Moritz Schlick's entry into the group, which led to the founding of the Vienna Circle
The Vienna Circle (german: Wiener Kreis) of Logical Empiricism was a group of philosophers and scientists drawn from the Natural science, natural and Social Sciences, social sciences, logic and mathematics who met regularly from 1924 to 1936 at th ...

, the group that was at the center of logical positivist thought in the 1920s. His most famous student was Kurt Gödel, whose Ph.D. thesis was completed in 1929. After Anschluss the fact that Hans Hahn had been of Jewish origin caused Gödel's difficulties with getting a position at The University of Vienna. Within the Vienna Circle, Hahn was also known (and controversial) for using his mathematical and philosophical work to study Extrasensory perception, psychic phenomena; according to Karl Menger he sometimes openly advocated further research into extrasensory perception while lecturing.
Hahn's contributions to mathematics include the Hahn–Banach theorem and (independently of Stefan Banach, Banach and Hugo Steinhaus, Steinhaus) the uniform boundedness principle. Other theorems include:
* the Hahn decomposition theorem;
* the Hahn embedding theorem;
* the Hahn–Kolmogorov theorem;
* the Space-filling curve, Hahn–Mazurkiewicz theorem;
* the Vitali–Hahn–Saks theorem.
Hahn authored the book : according to Arthur Rosenthal, "''... (it) formed a great advance in the Theory of Real functions and had a great influence on the further development of this theory''". He was also a co-author of the book ''Set Functions'', published in 1948 by Arthur Rosenthal, fourteen years after his death in Vienna in 1934.
In 1921 he received the Lieben Prize#Richard Lieben Prize, Richard Lieben Prize. In 1926 he was the president of the German Mathematical Society. In 1928 he was an Invited Speaker at the International Congress of Mathematicians in Bologna.
Publications

All his mathematical and philosophical works, except all books and all but one of his book reviews, are published in the three volumes , and of his "''Collected papers''". * (freely available at the Internet Archive). * * * *. *.See also

*Domain (mathematical analysis) *Mathematical analysisNotes

External links

* * {{DEFAULTSORT:Hahn, Hans 1879 births 1934 deaths Austrian Jews Scientists from Vienna 20th-century mathematicians Vienna Circle Topologists Set theorists Austrian agnostics Austrian mathematicians Jewish agnostics Jewish scientists Austro-Hungarian mathematicians Chernivtsi University faculty