Hans Hahn
Encyclopedia
Hans Hahn was an Austria
n mathematician who made contributions to functional analysis
, topology
, set theory
, the calculus of variations
, real analysis
, and order theory
.
as the son of a higher government official of the k.k. Telegraphen-Korrespondenz-Bureau (since 1946 named with the linguistically strange mixture "Austria Presse Agentur"), in 1898 he 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
, Munich
and Göttingen
. In 1902 he took his Ph.D. in Vienna.
He was appointed to the teaching staff (Habilitation
) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz
at Innsbruck
and some further years as a Privatdozent
in Vienna, he was nominated in 1909 Professor extraordinarius in Czernowitz
, 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, now in Bonn
. In 1917 he was nominated a regular Professor
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 a surgery.
He had married in 1909 Eleonore ("Lilly") Minor and they had a daughter, Nora (born 1910).
He was also interested in philosophy, and was part of a discussion group concerning Mach
's positivism
with Otto Neurath
and Phillip Frank prior to the First World War. In 1922, he helped arrange Moritz Schlick's
entry into the group, which led to the founding of the Vienna Circle
, 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.
Hahn's contributions to mathematics include the Hahn–Banach theorem
and (independently of Banach and Steinhaus
) the uniform boundedness principle
. Other theorems include:
Hahn was also a co-author of the book Set Functions. It was published in 1948, fourteen years after his death in Vienna in 1934.
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
n mathematician who made contributions to functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, and order theory
Order theory
Order theory is a branch of mathematics which investigates our 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 introduces the field and gives some basic definitions...
.
Biography
Born at ViennaVienna
Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...
as the son of a higher government official of the k.k. Telegraphen-Korrespondenz-Bureau (since 1946 named with the linguistically strange mixture "Austria Presse Agentur"), in 1898 he became a student at the Universität Wien
University of Vienna
The University of Vienna is a public university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world...
starting with a study of law. In 1899 he switched over to mathematics and spent some time at the universities of Strasbourg
Strasbourg
Strasbourg is the capital and principal city of the Alsace region in eastern France and is the official seat of the European Parliament. Located close to the border with Germany, it is the capital of the Bas-Rhin département. The city and the region of Alsace are historically German-speaking,...
, Munich
Munich
Munich The city's motto is "" . Before 2006, it was "Weltstadt mit Herz" . Its native name, , is derived from the Old High German Munichen, meaning "by the monks' place". The city's name derives from the monks of the Benedictine order who founded the city; hence the monk depicted on the city's coat...
and Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
. In 1902 he took his Ph.D. in Vienna.
He was appointed to the teaching staff (Habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz
Otto Stolz
Otto Stolz was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals. Born in Hall in Tirol, he studied in Innsbruck from 1860 and in Vienna from 1863, receiving his habilitation there in 1867...
at Innsbruck
Innsbruck
- Main sights :- Buildings :*Golden Roof*Kaiserliche Hofburg *Hofkirche with the cenotaph of Maximilian I, Holy Roman Emperor*Altes Landhaus...
and some further years as a Privatdozent
Privatdozent
Privatdozent or Private lecturer is a title conferred in some European university systems, especially in German-speaking countries, for someone who pursues an academic career and holds all formal qualifications to become a tenured university professor...
in Vienna, he was nominated in 1909 Professor extraordinarius in Czernowitz
Chernivtsi
Chernivtsi is the administrative center of Chernivtsi Oblast in southwestern Ukraine. The city is situated on the upper course of the River Prut, a tributary of the Danube, in the northern part of the historic region of Bukovina, which is currently divided between Romania and Ukraine...
, 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, now in Bonn
Bonn
Bonn is the 19th largest city in Germany. Located in the Cologne/Bonn Region, about 25 kilometres south of Cologne on the river Rhine in the State of North Rhine-Westphalia, it was the capital of West Germany from 1949 to 1990 and the official seat of government of united Germany from 1990 to 1999....
. In 1917 he was nominated a regular Professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
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 a surgery.
He had married in 1909 Eleonore ("Lilly") Minor and they had a daughter, Nora (born 1910).
He was also interested in philosophy, and was part of a discussion group concerning Mach
Ernst Mach
Ernst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...
's positivism
Logical positivism
Logical positivism is a philosophy that combines empiricism—the idea that observational evidence is indispensable for knowledge—with a version of rationalism incorporating mathematical and logico-linguistic constructs and deductions of epistemology.It may be considered as a type of analytic...
with Otto Neurath
Otto Neurath
Otto Neurath was an Austrian philosopher of science, sociologist, and political economist...
and Phillip Frank prior to the First World War. In 1922, he helped arrange Moritz Schlick's
Moritz Schlick
Friedrich Albert Moritz Schlick was a German philosopher, physicist and the founding father of logical positivism and the Vienna Circle.-Early life and works:...
entry into the group, which led to the founding of the Vienna Circle
Vienna Circle
The Vienna Circle was an association of philosophers gathered around the University of Vienna in 1922, chaired by Moritz Schlick, also known as the Ernst Mach Society in honour of Ernst Mach...
, the group that was at the center of logical positivist thought in the 1920s. His most famous student was Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
, whose Ph.D. thesis was completed in 1929.
Hahn's contributions to mathematics include the Hahn–Banach theorem
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
and (independently of Banach and Steinhaus
Hugo Steinhaus
Władysław Hugo Dionizy Steinhaus was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the University of Lwów, where he helped establish what later became known as the Lwów School of Mathematics...
) the uniform boundedness principle
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field...
. Other theorems include:
- the Hahn decomposition theoremHahn decomposition theoremIn mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space and a signed measure μ defined on the σ-algebra Σ, there exist two sets P and N in Σ such that:...
; - the Hahn embedding theoremHahn embedding theoremIn mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups....
; - the Hahn–Kolmogorov theorem;
- the Hahn–Mazurkiewicz theoremSpace-filling curveIn mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square...
; - the Vitali–Hahn–Saks theorem.
Hahn was also a co-author of the book Set Functions. It was published in 1948, fourteen years after his death in Vienna in 1934.