Rådström was interested in Hilbert's fifth problem on the analyticity of the continuous operation of topological groups. The solution of this problem by Andrew Gleason used constructions of subsets of topological vector spaces,, and inspired Rådström's research on set-valued analysis. He visited the Institute for Advanced Study in Princeton from 1948 to 1950, where he co-organized a seminar on convexity. Together with Olof Hanner, who, like Rådström, would earn his Ph.D. from Stockholm University in 1952, he improved Werner Fenchel's version of Carathéodory's lemma. In the 1950s, he obtained important results on convex sets. He proved the Rådström embedding theorem, which implies that the collection of all nonemptycompact convex subsets of a normed real vector-space can be isometrically embedded as a convex cone in a normed real vector-space. Under the embedding, the nonempty compact convex sets are mapped to points in the range 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 proved a variant of this theorem for locally convex topological vector spaces using the support function ; in Hörmander's approach, the range of the embedding was the Banach latticeL1, and the embedding was isotone. Rådström characterized the generators of continuous semigroups of sets as compact convex sets.
Students
Rådström's Ph.D. students included Per Enflo and Martin Ribe, both of whom wrote Ph.D. theses in functional analysis. In the uniform and Lipschitzcategories of topological vector spaces, Enflo's results concerned spaces with local convexity, especially Banach spaces. In 1970, Hans Rådström died of a heart attack. 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 Hilbert's fifth problem for complete, normed spaces. The Swedish functional analystEdgar Asplund, then Professor of Mathematics at Aarhus University 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 to topological vector spaces that lack local convexity.