Otto Hölder
Encyclopedia
Otto Ludwig Hölder was a German
mathematician
born in Stuttgart
.
Hölder first studied at the Polytechnikum (which today is the University of Stuttgart
) and then in 1877 went to Berlin
where he was a student of Leopold Kronecker
, Karl Weierstraß, and Ernst Kummer
.
He is famous for many things including: Hölder's inequality
, the Jordan–Hölder theorem, the theorem stating that every linearly ordered group
that satisfies an Archimedean property
is isomorphic to a subgroup of the additive group
of real number
s, the classification of simple group
s of order up to 200
, and Hölder's theorem
which implies that the Gamma function
satisfies no algebraic differential equation
. Another important notion related to his name is the Hölder condition
(or Hölder continuity) which is used in many areas of analysis
, including the theories of partial differential equations and function spaces.
In 1877, he entered the University of Berlin and took his doctorate from the University of Tübingen in 1882. The title of his doctoral thesis was "Beiträge zur Potentialtheorie" ("Contributions to potential theory
"). He worked at the University of Leipzig
from 1899 until his retirement.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
born in Stuttgart
Stuttgart
Stuttgart is the capital of the state of Baden-Württemberg in southern Germany. The sixth-largest city in Germany, Stuttgart has a population of 600,038 while the metropolitan area has a population of 5.3 million ....
.
Hölder first studied at the Polytechnikum (which today is the University of Stuttgart
University of Stuttgart
The University of Stuttgart is a university located in Stuttgart, Germany. It was founded in 1829 and is organized in 10 faculties....
) and then in 1877 went to Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
where he was a student of Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
, Karl Weierstraß, and Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
.
He is famous for many things including: Hölder's inequality
Hölder's inequality
In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces....
, the Jordan–Hölder theorem, the theorem stating that every linearly ordered group
Linearly ordered group
In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation "≤" is total...
that satisfies an Archimedean property
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...
is isomorphic to a subgroup of the additive group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, the classification of simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
s of order up to 200
200 (number)
200 is the natural number following 199 and preceding 201.The number appears in the Padovan sequence, preceded by 86, 114, 151 ....
, and Hölder's theorem
Hölder's theorem
In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. The result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.The theorem also generalizes to...
which implies that the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
satisfies no algebraic differential equation
Algebraic differential equation
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used....
. Another important notion related to his name is the Hölder condition
Hölder condition
In mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha , such that...
(or Hölder continuity) which is used in many areas of analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, including the theories of partial differential equations and function spaces.
In 1877, he entered the University of Berlin and took his doctorate from the University of Tübingen in 1882. The title of his doctoral thesis was "Beiträge zur Potentialtheorie" ("Contributions to potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
"). He worked at the University of Leipzig
University of Leipzig
The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...
from 1899 until his retirement.