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 ...

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 ...

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