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, structure, space, models, and change.
History
On ...
born in
Stuttgart
Stuttgart (; Swabian: ; ) is the capital and largest city of the German state of Baden-Württemberg. It is located on the Neckar river in a fertile valley known as the ''Stuttgarter Kessel'' (Stuttgart Cauldron) and lies an hour from the ...
.
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
The University of Stuttgart (german: Universität Stuttgart) is a leading research university located in Stuttgart, Germany. It was founded in 1829 and is organized into 10 faculties. It is one of the oldest technical universities in Germany wit ...
) and then in 1877 went to
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
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, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
,
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, 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
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 ...
.
[
In 1877, he entered the ]University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...
and took his doctorate from the University of Tübingen
The University of Tübingen, officially the Eberhard Karl University of Tübingen (german: Eberhard Karls Universität Tübingen; la, Universitas Eberhardina Carolina), is a public research university located in the city of Tübingen, Baden-Wü ...
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 two fundamental forces of nature known at the time, namely gravi ...
"). Following this, he went to the University of Leipzig
Leipzig University (german: Universität Leipzig), 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 Decemb ...
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, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
, both in 1884.
Academic career and later life
He was unable to get government approval for a faculty position in Göttingen
Göttingen (, , ; nds, Chöttingen) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. At the end of 2019, t ...
, and instead was offered a position as extraordinary professor at Tübingen
Tübingen (, , Swabian: ''Dibenga'') is a traditional university city in central Baden-Württemberg, Germany. It is situated south of the state capital, Stuttgart, and developed on both sides of the Neckar and Ammer rivers. about one in thr ...
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.
Life and career
Marius Sophu ...
as a full professor at the University of Leipzig
Leipzig University (german: Universität Leipzig), 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 Decemb ...
. 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
Ernst Otto Hölder (2 April 1901, Leipzig – 30 June 1990, Mainz) was a German mathematician who made contributions to partial differential equations and continuum mechanics.
Education and career
Hölder was born in Leipzig and studied at the ...
became another mathematician,[ and his daughter Irmgard married mathematician ]Aurel Wintner
Aurel Friedrich Wintner (8 April 1903 – 15 January 1958) was a mathematician noted for his research in mathematical analysis, number theory, differential equations and probability theory. He was one of the founders of probabilistic number theor ...
.
In 1933, Hölder signed the ''''.
Mathematical contributions
Holder's inequality, named for Hölder, was actually proven earlier by Leonard James Rogers
Leonard James Rogers Royal Society, FRS (30 March 1862 – 12 September 1933) was a British mathematician who was the first to discover the Rogers–Ramanujan identity and Hölder's inequality, and who introduced Rogers polynomials. The Roger ...
. 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
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier pr ...
, with some side conditions that were later removed by Jensen.
Hölder is also noted for many other theorems 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 natura ...
, the theorem stating that every linearly ordered group
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a:
* lef ...
that satisfies an 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 algebraic structures, such as ordered or normed groups, and fields.
The property, typical ...
is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...
to a subgroup of the additive group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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
__NOTOC__
Year 200 ( CC) was a leap year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Victorinus (or, less frequently, year 953 '' Ab ur ...
, 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 group \m ...
S6, and Hölder's theorem, which implies that the Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
satisfies no 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.
The intention is to in ...
. Another idea related to his name is the Hölder condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α > 0, such that
: , f(x) - f(y) , \leq C\, ...
(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 imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s and function spaces
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 vector ...
.
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
Leipzig University faculty
University of Stuttgart alumni
University of Tübingen faculty
University of Göttingen faculty