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Hans Hahn (September 27,1879 - July 24,1934) was an Austria
Austria

Austria , officially the Republic of Austria , is a landlocked country in Central Europe. It borders both Germany and the Czech Republic to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the west....
n mathematician who made contributions to functional analysis
Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
, topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, set theory
Set theory

Set theory is the branch of mathematics that studies Set , 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 functional , as opposed to ordinary calculus which deals with function . Such functionals can for example be formed as integrals involving an unknown function and its derivatives....
, real analysis
Real analysis

Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the Set of real numbers. In particular, it deals with the analytic function properties of real function and sequences, including convergence and limit s of sequences of real numbers, the calculus of the real numbers, and continu...
, and order theory
Order theory

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another....
. He was a student at the Technische Hochschule in Vienna.

He also studied in Strasbourg
Strasbourg

Strasbourg is the capital and principal city of the Alsace Regions of France in northeastern France. With 702,412 inhabitants in 2007, its metropolitan area is the Aire urbaine....
, Munich
Munich

Munich is the capital city of Bavaria, Germany. Munich is located on the River Isar north of the Northern Limestone Alps. Munich is the third largest city in Germany, after Berlin and Hamburg....
 and Göttingen
Göttingen

G?ttingen is a college town in Lower Saxony, Germany. It is the Capital of the district of G?ttingen . The Leine river runs through the town. In 2006 the population was 129,686....
. He was appointed to the teaching staff in Vienna in 1905 and became professor of mathematics there in 1921. In session 1905-06 Hahn substituted for Otto Stolz
Otto Stolz

Otto Stolz was an Austrians mathematician noted for his work on mathematical analysis and infinitesimals. His name lives on in the Stolz-Ces?ro theorem....
 at Innsbruck
Innsbruck

Innsbruck is the Capital of the federal state of Tyrol in western Austria. It is located in the Inn River Valley at the junction with the Wipptal , which provides access to the Brenner Pass, some 30 km south of Innsbruck....
.

He was also interested in philosophy, and was part of a discussion group concerning Mach's positivism with Otto Neurath
Otto Neurath

Otto Neurath was an Austrian philosophy of science, sociology, and political economy. Before he was forced to flee his native country for Great Britain in the wake of the Nazism occupation, Neurath was one of the leading figures of the Vienna Circle....
 and Phillip Frank prior to the first world war.






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Hans Hahn (September 27,1879 - July 24,1934) was an Austria
Austria

Austria , officially the Republic of Austria , is a landlocked country in Central Europe. It borders both Germany and the Czech Republic to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the west....
n mathematician who made contributions to functional analysis
Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
, topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, set theory
Set theory

Set theory is the branch of mathematics that studies Set , 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 functional , as opposed to ordinary calculus which deals with function . Such functionals can for example be formed as integrals involving an unknown function and its derivatives....
, real analysis
Real analysis

Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the Set of real numbers. In particular, it deals with the analytic function properties of real function and sequences, including convergence and limit s of sequences of real numbers, the calculus of the real numbers, and continu...
, and order theory
Order theory

Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another....
. He was a student at the Technische Hochschule in Vienna.

He also studied in Strasbourg
Strasbourg

Strasbourg is the capital and principal city of the Alsace Regions of France in northeastern France. With 702,412 inhabitants in 2007, its metropolitan area is the Aire urbaine....
, Munich
Munich

Munich is the capital city of Bavaria, Germany. Munich is located on the River Isar north of the Northern Limestone Alps. Munich is the third largest city in Germany, after Berlin and Hamburg....
 and Göttingen
Göttingen

G?ttingen is a college town in Lower Saxony, Germany. It is the Capital of the district of G?ttingen . The Leine river runs through the town. In 2006 the population was 129,686....
. He was appointed to the teaching staff in Vienna in 1905 and became professor of mathematics there in 1921. In session 1905-06 Hahn substituted for Otto Stolz
Otto Stolz

Otto Stolz was an Austrians mathematician noted for his work on mathematical analysis and infinitesimals. His name lives on in the Stolz-Ces?ro theorem....
 at Innsbruck
Innsbruck

Innsbruck is the Capital of the federal state of Tyrol in western Austria. It is located in the Inn River Valley at the junction with the Wipptal , which provides access to the Brenner Pass, some 30 km south of Innsbruck....
.

He was also interested in philosophy, and was part of a discussion group concerning Mach's positivism with Otto Neurath
Otto Neurath

Otto Neurath was an Austrian philosophy of science, sociology, and political economy. Before he was forced to flee his native country for Great Britain in the wake of the Nazism occupation, Neurath was one of the leading figures of the Vienna Circle....
 and Phillip Frank prior to the first world war. In 1922, he helped arrange Moritz Schlick's
Moritz Schlick

Moritz Schlick was a Germany philosopher and the founding father of logical positivism and the Vienna Circle....
 entry into the group, which led to the founding of the Vienna Circle
Vienna Circle

The Vienna Circle was a group of philosophers who gathered around Moritz Schlick when he was called to the Vienna University in 1922, organized in a philosophical association, of which Schlick was chairman, named 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 G?del was an Austrian-United States logician, mathematician and philosopher. One of the most significant logicians of all time, G?del made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A....
, whose Ph.D. thesis was completed in 1929.

Hahn's contributions to mathematics include the Hahn-Banach theorem and (independently of Banach and Steinhaus
Hugo Steinhaus

Wladyslaw Hugo Dionizy Steinhaus was a Poland mathematician and educator....
) 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 theorem
    Hahn decomposition theorem

    In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a sigma-algebra 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 theorem
    Hahn embedding theorem

    In 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 totally ordered abelian groups....
    ;
  • the Hahn-Kolmogorov theorem
    Hahn-Kolmogorov theorem

    In mathematics, the Hahn?Kolmogorov theorem characterizes when a finitely additive function with negative and positive numbers values can be extended to a bona fide measure ....
    ;
  • the Hahn-Mazurkiewicz theorem
    Space-filling curve

    In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square . Because Giuseppe Peano was the first to discover one, space-filling curves in the Plane are commonly called Peano curves....
    ;
  • the Vitali-Hahn-Saks theorem.


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