Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
born in
Stuttgart
Stuttgart (; ; Swabian German, Swabian: ; Alemannic German, Alemannic: ; Italian language, Italian: ; ) is the capital city, capital and List of cities in Baden-Württemberg by population, largest city of the States of Germany, German state of ...
.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christian Gottlieb Hölder (1776–1847); his two brothers also became professors. He first studied at the ''Polytechnikum'' (which today is the
University of Stuttgart) and then in 1877 went to
Berlin
Berlin ( ; ) is the Capital of Germany, capital and largest city of Germany, by both area and List of cities in Germany by population, population. With 3.7 million inhabitants, it has the List of cities in the European Union by population withi ...
where he was a student of
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker
as having said, ...
,
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
, and
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) 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 h ...
.
[
In 1877, he entered the ]University of Berlin
The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany.
The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...
and took his doctorate from the University of Tübingen
The University of Tübingen, officially the Eberhard Karl University of Tübingen (; ), is a public research university located in the city of Tübingen, Baden-Württemberg, Germany.
The University of Tübingen is one of eleven German Excellenc ...
in 1882. The title of his doctoral thesis was "Beiträge zur Potentialtheorie" ("Contributions to potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...
"). Following this, he went to the University of Leipzig
Leipzig University (), in Leipzig in Saxony, Germany, is one of the world's oldest universities and the second-oldest university (by consecutive years of existence) in Germany. The university was founded on 2 December 1409 by Frederick I, Electo ...
but was unable to habilitate there, instead earning a second doctorate and habilitation at the University of Göttingen
The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
, both in 1884.
Academic career and later life
He was unable to get government approval for a faculty position in Göttingen, and instead was offered a position as extraordinary professor at Tübingen in 1889. Temporary mental incapacitation delayed his acceptance but he began working there in 1890. In 1899, he took the former chair of Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial cont ...
as a full professor at the University of Leipzig. There he served as dean from 1912 to 1913, and as rector in 1918.[
He married Helene, the daughter of a bank director and politician, in 1899. They had two sons and two daughters. His son Ernst Hölder became another mathematician,][ and his daughter Irmgard married mathematician Aurel Wintner.
In 1933, Hölder signed the '' Vow of allegiance of the Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State''.
]
Mathematical contributions
Holder's inequality, named for Hölder, was actually proven earlier by Leonard James Rogers. It is named for a paper in which Hölder, citing Rogers, reproves it; in turn, the same paper includes a proof of what is now called Jensen's inequality, with some side conditions that were later removed by Jensen.
Hölder is also noted for many other theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s including the Jordan–Hölder theorem
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many na ...
, the theorem stating that every linearly ordered group
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group (mathematics), group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G ...
that satisfies an Archimedean property
In abstract algebra and mathematical analysis, analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, Italy, Syracuse, is a property held by some algebraic structures, such as ordered or normed g ...
is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to a subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of the additive group of real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, the classification of simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
s of order up to 200, the anomalous outer automorphisms of the symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
''S''6, and Hölder's theorem, which implies that the Gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
satisfies no algebraic differential equation. Another idea related to his name is the Hölder condition
In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that
, f(x) - f(y) , \leq C\, x - y\, ^
for all and in the do ...
(or Hölder continuity), which is used in many areas of analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, including the theories of partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s and function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
s.
References
{{DEFAULTSORT:Holder, Otto
Group theorists
19th-century German mathematicians
20th-century German mathematicians
1859 births
1937 deaths
Scientists from Stuttgart
Humboldt University of Berlin alumni
University of Tübingen alumni
Academic staff of Leipzig University
University of Stuttgart alumni
Academic staff of the University of Tübingen
Academic staff of the University of Göttingen
Presidents of the German Mathematical Society