Hans Vilhem Rådström (1919–1970) was a Swedish mathematician who worked on

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

continuous group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

set-valued analysis A set-valued function (or correspondence) is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including optimizatio ...

, and

game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

. From 1952, he was ''lektor'' (

assistant professor Assistant Professor is an academic rank just below the rank of an associate professor used in universities or colleges, mainly in the United States and Canada. Overview This position is generally taken after earning a doctoral degree and general ...

) at

Stockholm University Stockholm University ( sv, Stockholms universitet) is a public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, so ...

, and from 1969, he was Professor of Applied Mathematics at

Linköping University Linköping University (, LiU) is a public research university based in Linköping, Sweden. Originally established in 1969, it was granted full university status in 1975 and is one of Sweden's largest academic institutions. The university has fou ...

Early life

Hans Rådström was the son of the writer and editor Karl Johan Rådström, and the elder brother of the writer and journalist

Pär Rådström Pär Kristian Rådström (29 August, 1925 – 29 August, 1963) was a Swedish writer and journalist. He also wrote popular songs and was a radio personality. His novel, ''Ärans portar'' (''The Gates of Glory''), published in 1954, was his first ...

. Rådström studied mathematics and obtained his Ph.D. under the joint supervision of

Torsten Carleman Torsten Carleman (8 July 1892, Visseltofta, Osby Municipality – 11 January 1949, Stockholm), born Tage Gillis Torsten Carleman, was a Swedish mathematician, known for his results in classical analysis and its applications. As the director of th ...

and

Fritz Carlson Fritz David Carlson (23 July 1888 – 28 November 1952) was a Swedish mathematician. After the death of Torsten Carleman, he headed the Mittag-Leffler Institute. Carlson's contributions to analysis include Carlson's theorem, the Polyá–Carl ...

. His early work pertained to the theory of functions of a complex variable, particularly,

complex dynamics Complex dynamics is the study of dynamical systems defined by Iterated function, iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Mo ...

. He was appointed ''lektor'' (

) at Stockholm University in 1952. Later, he was associated with the

Royal Institute of Technology The KTH Royal Institute of Technology ( sv, Kungliga Tekniska högskolan, lit=Royal Institute of Technology), abbreviated KTH, is a public research university in Stockholm, Sweden. KTH conducts research and education in engineering and technolo ...

in Stockholm. In 1952 he became co-editor of the Scandinavian popular-mathematics journal ''Nordisk Matematisk Tidskrift''. He also edited the Swedish edition of ''The Scientific American Book of Mathematical Puzzles and Diversions'', a

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

book by

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...

Set-valued analysis

Rådström was interested in

Hilbert's fifth problem Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathemat ...

on the analyticity of the continuous operation of

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...

s. The solution of this problem by

Andrew Gleason Andrew Mattei Gleason (19212008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching at ...

used constructions of

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s of

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

s, (rather than simply

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

s), and inspired Rådström's research on

. He visited the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

(IAS) in Princeton from 1948 to 1950, where he co-organized a seminar on convexity. Together with

Olof Hanner Olof Hanner (7 December 1922 in Stockholm – 19 September 2015 in Gothenburg) was a Swedish mathematician. Education and career Hanner earned his Ph.D. from Stockholm University in 1952. He was a professor at the University of Gothenburg from 196 ...

, who, like Rådström, would earn his Ph.D. from Stockholm University in 1952, he improved

Werner Fenchel Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theor ...

's version of

Carathéodory's lemma In mathematics, Nevanlinna's criterion in complex analysis, proved in 1920 by the Finnish mathematician Rolf Nevanlinna, characterizes holomorphic univalent functions on the unit disk which are starlike. Nevanlinna used this criterion to prove ...

. In the 1950s, he obtained important results on

s. He proved the ''Rådström embedding theorem'', which implies that the collection of all

nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...

compact convex subsets of a normed real vector-space (endowed with the

Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric ...

) can be isometrically embedded as a

convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . ...

in a normed real vector-space. Under the embedding, the nonempty compact convex sets are mapped to

s in the

range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...

space. In Rådström's construction, this embedding is additive and positively homogeneous. Rådström's approach used ideas from the theory of topological semi-groups. Later,

Lars Hörmander Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Medal ...

proved a variant of this theorem for

locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...

s using the

support function In mathematics, the support function ''h'A'' of a non-empty closed convex set ''A'' in \mathbb^n describes the (signed) distances of supporting hyperplanes of ''A'' from the origin. The support function is a convex function on \mathbb^n. Any n ...

(of

convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of s ...

); in Hörmander's approach, the range of the embedding was the Banach

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornam ...

''L''₁, and the embedding was

isotone Two nuclides are isotones if they have the same neutron number ''N'', but different proton number ''Z''. For example, boron-12 and carbon-13 nuclei both contain 7 neutrons, and so are isotones. Similarly, 36S, 37Cl, 38Ar, 39K, and 40Ca nuclei ...

. : Rådström characterized the generators of continuous semigroups of sets as

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

convex sets.

Students

Rådström's Ph.D. students included

Per Enflo Per H. Enflo (; born 20 May 1944) is a Swedish mathematician working primarily in functional analysis, a field in which he solved problems that had been considered fundamental. Three of these problems had been open for more than forty years: * The ...

and Martin Ribe, both of whom wrote Ph.D. theses in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...

. In the

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

and Lipschitz

categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...

s, Enflo's results concerned spaces with local convexity, especially

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

s. In 1970, Hans Rådström died of a

heart attack A myocardial infarction (MI), commonly known as a heart attack, occurs when blood flow decreases or stops to the coronary artery of the heart, causing damage to the heart muscle. The most common symptom is chest pain or discomfort which may tr ...

. Enflo supervised one of Rådström's Linköping students, Lars-Erik Andersson, from 1970–1971, helping him with his 1972 thesis, ''On connected subgroups of Banach spaces'', on

for complete, normed spaces. The Swedish functional analyst

Edgar Asplund Edgar is a commonly used English given name, from an Anglo-Saxon name ''Eadgar'' (composed of '' ead'' "rich, prosperous" and ''gar'' "spear"). Like most Anglo-Saxon names, it fell out of use by the later medieval period; it was, however, rev ...

, then Professor of Mathematics at

Aarhus University Aarhus University ( da, Aarhus Universitet, abbreviated AU) is a public research university with its main campus located in Aarhus, Denmark. It is the second largest and second oldest university in Denmark. The university is part of the Coimbra Gr ...

in Denmark, assisted Ribe as supervisor of his 1972 thesis, before dying of cancer in 1974. Ribe's results concerned topological vector spaces without assuming local convexity; Ribe constructed a counter-example to naive extensions of the

Hahn–Banach theorem The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...

to topological vector spaces that lack local convexity.

References

External links

*
Hans Rådström
at

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

{{DEFAULTSORT:Radstrom, Hans 1919 births 1970 deaths Functional analysts Mathematical analysts 20th-century Swedish mathematicians