Hans Rådström: Difference between revisions

Hans Rådström
Hans Rådström
Born	1919
Died	1970
Citizenship	Sweden
Alma mater	Stockholm University
Known for	Rådström isometric embedding of convex subsets in the positive cone of the Lebesgue space of absolutely integrable functions; Rådström characterization of convex sets as generators of continuous semigroups of subsets
	Scientific career
Fields	Functional equations, set-valued analysis
Institutions	Institute for Advanced Study, Princeton University; Stockholm University; Linköping University
Doctoral advisor	Torsten Carleman, Fritz Carlson
Doctoral students	Per Enflo

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Hans Vilhem Rådström (1919–1970) was a Swedish mathematician who worked on complex analysis, continuous groups, convex sets, set-valued analysis, and game theory. From 1952, he was lektor (assistant professor) at Stockholm University,^[1] and from 1969, he was Professor of Applied Mathematics at Linköping University.^[2]

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.

Rådström studied mathematics and obtained his Ph.D. under the joint supervision of Torsten Carleman and Fritz Carlson. His early work pertained to the theory of functions of a complex variable, particularly, complex dynamics. He was appointed lektor (assistant professor) at Stockholm University in 1952.^[1] Later, he was associated with the Royal Institute of Technology in Stockholm.

In 1952 he became co-editor of the Scandinavian popular-mathematics journal Nordisk Matematisk Tidskrift.^[3] He also edited the Swedish edition of The Scientific American Book of Mathematical Puzzles and Diversions, a recreational mathematics book by Martin Gardner.^[4]

Set-valued analysis

Lars Hörmander (pictured) proved a variant of Rådström's embedding theorem using support functions.

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,^[5] (rather than simply points), and inspired Rådström's research on set-valued analysis.

He visited the Institute for Advanced Study (IAS) in Princeton from 1948 to 1950,^[6] where he co-organized a seminar on convexity.^[7] 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.^[8]

In the 1950s, he obtained important results on convex sets. He proved the Rådström embedding theorem, which implies that the collection of all nonempty compact convex subsets of a normed real vector-space (endowed with the Hausdorff distance) 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.^[9] Rådström's approach used ideas from the theory of topological semi-groups.^[10] Later, Lars Hörmander proved a variant of this theorem for locally convex topological vector spaces using the support function (of convex analysis); in Hörmander's approach, the range of the embedding was the Banach lattice L₁, and the embedding was isotone.^[9]^[10]^[11]

Rådström characterized the generators of continuous semigroups of sets as compact convex sets.^[12]

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 Lipschitz categories of topological vector spaces, Enflo's results^[13] concerned spaces with local convexity, especially Banach spaces.^[14]^[15]

In 1970,^[16] Hans Rådström died of a heart attack.^[17] Enflo supervised one of Rådström's Linköping students, Lars-Erik Andersson, from 1970–1971, helping him with his 1972 thesis,^[17] On connected subgroups of Banach spaces, on Hilbert's fifth problem for complete, normed spaces. The Swedish functional analyst Edgar Asplund, then Professor of Mathematics at Aarhus University in Denmark, assisted Ribe as supervisor of his 1972 thesis,^[18] before dying of cancer in 1974.^[19] Ribe's results concerned topological vector spaces without assuming local convexity;^[14] Ribe constructed a counter-example to naive extensions of the Hahn–Banach theorem to topological vector spaces that lack local convexity.^[20]

References

^ ^a ^b "Notes". Bulletin of the American Mathematical Society. 58 (6): 683–692. 1952. doi:10.1090/s0002-9904-1952-09670-1.
^ "LiTH—från plan till verklighet, Åke Björck" (PDF) (in Swedish). Linköping University. 27 January 2010. Archived from the original (PDF) on 6 April 2012. Retrieved 29 December 2011. (Webpage of Professor Åke Björck at Linköping University)
^ Branner, Bodil (2003). "On the Foundation of Mathematica Scandinavica". Mathematica Scandinavica. 93: 5–18. doi:10.7146/math.scand.a-14409.
^ Gardner, M. (1961). Rolig Matematik: Tankenötter och Problem, Andra Samlingen. Stockholm: Natur & Kultur., see "library card". Sollentuna library. Archived from the original on 2019-06-24. Retrieved 2018-10-19.
^ Gleason, Andrew (1952). "One-parameter subgroups and Hilbert's fifth problem". Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, 1950. Vol. 2. Providence, Rhode Island: American Mathematical Society. pp. 451–452.
^ "IAS Past Member Profile". Institute for Advanced Study. 2011. Retrieved 16 October 2021.
^ Bateman, P. T.; Rådström, Hans; Hanner, Olaf; Macbeath, A. M.; Rogers, C. A.; Pettis, B. J.; Klee, V. L. Seminar on convex sets, 1949–1950. Princeton, N. J.: The Institute for Advanced Study. MR 0064421.
^ Reay, John R. (1965). "Generalizations of a theorem of Carathéodory". Mem. Amer. Math. Soc. (Doctoral thesis). 54. Mathematics Department, University of Washington. MR 0188891.
^ ^a ^b Schneider (1993, Notes for section 1.8 (pp. 56–61, especially 57–58)): Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490. doi:10.1017/CBO9780511526282. ISBN 978-0-521-35220-8. MR 1216521.
^ ^a ^b Schmidt, Klaus D (March 1986). "Embedding theorems for classes of convex sets". Acta Applicandae Mathematicae. 5 (3): 209–237. doi:10.1007/BF00047343. S2CID 123172020.
^ Hörmander, Lars (1994). Notions of convexity. Progress in Mathematics. Vol. 127. Boston, MA: Birkhäuser Boston, Inc. ISBN 978-0-8176-3799-6. MR 1301332.
^ Hilgert, Joachim; Hofmann, Karl Heinrich; Lawson, Jimmie D. (1989). Lie groups, convex cones, and semigroups. Oxford Mathematical Monographs. Oxford University Press. ISBN 978-0-19-853569-0. LCCN 89009289.
^ Enflo, Per (1970). Investigations on Hilbert's fifth problem for non locally compact groups (doctoral thesis). Stockholm University.
^ ^a ^b Lindensrauss, Joram; Benyamini, Yoav. Geometric nonlinear functional analysis. Colloquium publications. Vol. 48. American Mathematical Society.
^ Matoušek, Jiří (2002). Lectures on Discrete Geometry. Graduate Texts in Mathematics. Springer-Verlag. ISBN 978-0-387-95373-1.
^ Kiselman (2010, p. 1436): Kiselman, Christer O. (2010). "Inverses and quotients of mappings between ordered sets". Image and Vision Computing. 28 (10): 1429–1442. doi:10.1016/j.imavis.2009.06.014.
^ ^a ^b Enflo, Per (25 April 2011). "Personal notes, in my own words". perenflo.com. Archived from the original on 26 April 2012. Retrieved 13 December 2011.
^ Acknowledgement in Ribe, Martin (1972). On Spaces Which Are Not Supposed to be Locally Convex (doctoral thesis). Linköping: Högsk.
^ Borwein, Jonathan M. (2007). "Asplund decompositions of monotone operators". ESAIM Proc. 17: 19–25. doi:10.1051/proc:071703. MR 2362689.
^ Kalton, Nigel J.; Peck, N. Tenney; Roberts, James W. (1984). An F-space sampler. London Mathematical Society Lecture Note Series. Vol. 89. Cambridge: Cambridge University Press. pp. xii+240. doi:10.1017/CBO9780511662447. ISBN 978-0-521-27585-9. MR 0808777.

External links

Hans Rådström at the Mathematics Genealogy Project
Hans Rådström at Mathematical Reviews

[notes-1] "Notes". Bulletin of the American Mathematical Society. 58 (6): 683–692. 1952. doi:10.1090/s0002-9904-1952-09670-1.

[2] "LiTH—från plan till verklighet, Åke Björck" (PDF) (in Swedish). Linköping University. 27 January 2010. Archived from the original (PDF) on 6 April 2012. Retrieved 29 December 2011. (Webpage of Professor Åke Björck at Linköping University)

[3] Branner, Bodil (2003). "On the Foundation of Mathematica Scandinavica". Mathematica Scandinavica. 93: 5–18. doi:10.7146/math.scand.a-14409.

[4] Gardner, M. (1961). Rolig Matematik: Tankenötter och Problem, Andra Samlingen. Stockholm: Natur & Kultur., see "library card". Sollentuna library. Archived from the original on 2019-06-24. Retrieved 2018-10-19.

[5] Gleason, Andrew (1952). "One-parameter subgroups and Hilbert's fifth problem". Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, 1950. Vol. 2. Providence, Rhode Island: American Mathematical Society. pp. 451–452.

[6] "IAS Past Member Profile". Institute for Advanced Study. 2011. Retrieved 16 October 2021.

[7] Bateman, P. T.; Rådström, Hans; Hanner, Olaf; Macbeath, A. M.; Rogers, C. A.; Pettis, B. J.; Klee, V. L. Seminar on convex sets, 1949–1950. Princeton, N. J.: The Institute for Advanced Study. MR 0064421.

[8] Reay, John R. (1965). "Generalizations of a theorem of Carathéodory". Mem. Amer. Math. Soc. (Doctoral thesis). 54. Mathematics Department, University of Washington. MR 0188891.

[Schneider-9] Schneider (1993, Notes for section 1.8 (pp. 56–61, especially 57–58)): Schneider, Rolf (1993). Convex bodies: The Brunn–Minkowski theory. Encyclopedia of mathematics and its applications. Vol. 44. Cambridge: Cambridge University Press. pp. xiv+490. doi:10.1017/CBO9780511526282. ISBN 978-0-521-35220-8. MR 1216521.

[Schmidt-10] Schmidt, Klaus D (March 1986). "Embedding theorems for classes of convex sets". Acta Applicandae Mathematicae. 5 (3): 209–237. doi:10.1007/BF00047343. S2CID 123172020.

[11] Hörmander, Lars (1994). Notions of convexity. Progress in Mathematics. Vol. 127. Boston, MA: Birkhäuser Boston, Inc. ISBN 978-0-8176-3799-6. MR 1301332.

[12] Hilgert, Joachim; Hofmann, Karl Heinrich; Lawson, Jimmie D. (1989). Lie groups, convex cones, and semigroups. Oxford Mathematical Monographs. Oxford University Press. ISBN 978-0-19-853569-0. LCCN 89009289.

[13] Enflo, Per (1970). Investigations on Hilbert's fifth problem for non locally compact groups (doctoral thesis). Stockholm University.

[GNFA-14] Lindensrauss, Joram; Benyamini, Yoav. Geometric nonlinear functional analysis. Colloquium publications. Vol. 48. American Mathematical Society.

[LDG-15] Matoušek, Jiří (2002). Lectures on Discrete Geometry. Graduate Texts in Mathematics. Springer-Verlag. ISBN 978-0-387-95373-1.

[16] Kiselman (2010, p. 1436): Kiselman, Christer O. (2010). "Inverses and quotients of mappings between ordered sets". Image and Vision Computing. 28 (10): 1429–1442. doi:10.1016/j.imavis.2009.06.014.

[EnfloWords-17] Enflo, Per (25 April 2011). "Personal notes, in my own words". perenflo.com. Archived from the original on 26 April 2012. Retrieved 13 December 2011.

[18] Acknowledgement in Ribe, Martin (1972). On Spaces Which Are Not Supposed to be Locally Convex (doctoral thesis). Linköping: Högsk.

[19] Borwein, Jonathan M. (2007). "Asplund decompositions of monotone operators". ESAIM Proc. 17: 19–25. doi:10.1051/proc:071703. MR 2362689.

[20] Kalton, Nigel J.; Peck, N. Tenney; Roberts, James W. (1984). An F-space sampler. London Mathematical Society Lecture Note Series. Vol. 89. Cambridge: Cambridge University Press. pp. xii+240. doi:10.1017/CBO9780511662447. ISBN 978-0-521-27585-9. MR 0808777.

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@@ Line 1: / Line 1: @@
+{{short description|Swedish mathematician}}
 {{Infobox scientist
-|name              = Hans Rådström
+| name              = Hans Rådström
-|image             =
+| image             =
-|image_size        =
+| image_size        =
-|caption           =
+| caption           =
-|birth_date        = 1919
+| birth_date        = 1919
-|birth_place       =
+| birth_place       =
-|death_date        = 1970
+| death_date        = 1970
-|death_place       =
+| death_place       =
-|residence         =
+| residence         =
-|citizenship       = Sweden
+| citizenship       = Sweden
-|nationality       =
+| nationality       =
-|ethnicity         =
+| ethnicity         =
-|fields            = [[Functional equation]]s, [[set-valued analysis]]
+| fields            = [[Functional equation]]s, [[set-valued analysis]]
-|workplaces        = [[Institute for Advanced Studies]], [[Princeton University]]; [[Stockholm University]]; [[Linköping University]]
+| workplaces        = [[Institute for Advanced Study]], [[Princeton University]]; [[Stockholm University]]; [[Linköping University]]
-|alma_mater        = [[Stockholm University]]
+| alma_mater        = [[Stockholm University]]
-|doctoral_advisor  = [[Torsten Carleman]], [[Fritz Carlson]]
+| doctoral_advisor  = [[Torsten Carleman]], [[Fritz Carlson]]
-|academic_advisors =
+| academic_advisors =
-|doctoral_students = [[Per Enflo]]
+| doctoral_students = [[Per Enflo]]
-|notable_students  =
+| notable_students  =
-|known_for         = Rådström isometric embedding of convex subsets in the positive [[convex cone|cone]] of the [[Lp space|Lebesgue space]] of [[absolutely integrable function]]s;  Rådström characterization of convex sets as generators of continuous semigroups of subsets
+| known_for         = Rådström isometric embedding of convex subsets in the positive [[convex cone|cone]] of the [[Lp space|Lebesgue space]] of [[absolutely integrable function]]s;  Rådström characterization of convex sets as generators of continuous semigroups of subsets
-|influences        = [[Werner Fenchel]] <br /> [[Andrew Gleason]]
+| awards            =
+| religion          =
-|influenced        = [[Karl Johan Åström]]<ref>{{cite web |url=http://www.tekniskamuseet.se/download/18.6aa228912529fe96108000358/3_Karl_Johan_%C3%85str%C3%B6m.pdf |title=Karl Johan Åström: En intervju av Per Lundin |language=Swedish |trans-title=Karl Johan Åström: An Interview with Per Lundin |date=3 October 2007 |accessdate=29 December 2011 |publisher=teknishkamuseet.se}}</ref>
-|awards            =
+| signature         = <!--(filename only)-->
-|religion          =
+| footnotes         =
-|signature         =  <!--(filename only)-->
-|footnotes         =
 }}
-'''Hans Vilhem Rådström''' (1919&ndash;1970) was a Swedish mathematician who worked on [[complex analysis]], [[Hilbert's fifth problem|continuous group]]s, [[convex set]]s, [[set-valued analysis]], and [[game theory]]. From 1952, he was [[academic rank in Sweden#Lektor|''lektor'']] ([[assistant professor]]) at [[Stockholm University]],<ref name=notes>{{cite journal|title=Notes|journal=[[Bulletin of the American Mathematical Society]]|volume=58|issue=6|year=1952|pages=683&ndash;692|url=http://projecteuclid.org/euclid.bams/1183517444|doi=10.1090/s0002-9904-1952-09670-1 }}</ref> and from 1969, he was Professor of Applied Mathematics at [[Linköping University]].<ref>{{cite web |url=http://www.mai.liu.se/~akbjo/collqmai.pdf |title=LiTH—från plan till verklighet, Åke Björck |language=Swedish |date=27 January 2010 |accessdate=29 December 2011 |publisher=[[Linköping University]] |deadurl=yes |archiveurl=https://web.archive.org/web/20120406081320/http://www.mai.liu.se/~akbjo/collqmai.pdf |archivedate=6 April 2012 |df= }} (Webpage of Professor Åke Björck at Linköping University)</ref>
+'''Hans Vilhem Rådström''' (1919&ndash;1970) was a Swedish mathematician who worked on [[complex analysis]], [[Hilbert's fifth problem|continuous group]]s, [[convex set]]s, [[set-valued analysis]], and [[game theory]]. From 1952, he was [[academic rank in Sweden#Lektor|''lektor'']] ([[assistant professor]]) at [[Stockholm University]],<ref name=notes>{{cite journal|title=Notes|journal=[[Bulletin of the American Mathematical Society]]|volume=58|issue=6|year=1952|pages=683&ndash;692|url=http://projecteuclid.org/euclid.bams/1183517444|doi=10.1090/s0002-9904-1952-09670-1 |doi-access=free}}</ref> and from 1969, he was Professor of Applied Mathematics at [[Linköping University]].<ref>{{cite web |url=http://www.mai.liu.se/~akbjo/collqmai.pdf |title=LiTH—från plan till verklighet, Åke Björck |language=sv |date=27 January 2010 |access-date=29 December 2011 |publisher=[[Linköping University]] |url-status=dead |archive-url=https://web.archive.org/web/20120406081320/http://www.mai.liu.se/~akbjo/collqmai.pdf |archive-date=6 April 2012 }} (Webpage of Professor Åke Björck at Linköping University)</ref>
 ==Early life==
@@ Line 35: / Line 34: @@
 Rådström studied mathematics and obtained his Ph.D. under the joint supervision of [[Torsten Carleman]] and [[Fritz Carlson]].  His early work pertained to the  [[theory of functions of a complex variable]], particularly, [[complex dynamics]]. He was appointed ''lektor'' ([[assistant professor]]) at Stockholm University in 1952.<ref name=notes/>  Later, he was associated with the [[Royal Institute of Technology]] in Stockholm.
-In 1952 he became co-editor of the Scandinavian popular-mathematics journal ''Nordisk Matematisk Tidskrift''.<ref>{{cite journal|last=Branner |first=Bodil|authorlink= Bodil Branner |title=On the Foundation of Mathematica Scandinavica|url=http://www.matilde.mathematics.dk/arkiv/M17/Branner_Math_Scand.pdf |publisher=matilde.mathematics.dk|journal=Mathematica Scandinavica |volume=93 |year=2003 |pages=5&ndash;18}}</ref> He also edited the Swedish edition of ''The Scientific American Book of Mathematical Puzzles and Diversions'', a [[recreational mathematics]] book by [[Martin Gardner]].<ref>{{cite book|last=Gardner|first=M.|author-link=Martin Gardner|title=Rolig Matematik: Tankenötter och Problem, Andra Samlingen|publisher=[[Natur & Kultur]]|location=Stockholm|year=1961}}, see {{cite web|url=https://www.bibliotekenisollentuna.se/web/arena/skaffa-bibliotekskort |title=library card|publisher=Sollentuna library }}</ref>
+In 1952 he became co-editor of the Scandinavian popular-mathematics journal ''Nordisk Matematisk Tidskrift''.<ref>{{cite journal |last=Branner |first=Bodil |author-link=Bodil Branner |title=On the Foundation of Mathematica Scandinavica |journal=Mathematica Scandinavica |volume=93 |year=2003 |pages=5&ndash;18 |doi=10.7146/math.scand.a-14409 |doi-access=free }}</ref> He also edited the Swedish edition of ''The Scientific American Book of Mathematical Puzzles and Diversions'', a [[recreational mathematics]] book by [[Martin Gardner]].<ref>{{cite book|last=Gardner|first=M.|author-link=Martin Gardner|title=Rolig Matematik: Tankenötter och Problem, Andra Samlingen|publisher=[[Natur & Kultur]]|location=Stockholm|year=1961}}, see {{cite web |url=https://www.bibliotekenisollentuna.se/web/arena/skaffa-bibliotekskort |title=library card |publisher=Sollentuna library |access-date=2018-10-19 |archive-date=2019-06-24 |archive-url=https://web.archive.org/web/20190624214616/https://www.bibliotekenisollentuna.se/web/arena/skaffa-bibliotekskort |url-status=dead }}</ref>
 ==Set-valued analysis==
 [[Image:Lars Hörmander.jpg|thumb|right|alt=Headshot of Lars Hörmander|[[Lars Hörmander]] ''(pictured)'' proved a variant of Rådström's embedding theorem using [[support function]]s.]]
-[[Image:Per Enflo.jpg|right|thumb|headshot of Per Enflo (Berkeley 1972)|[[Per Enflo]] ''(pictured)'' wrote his doctoral thesis under the supervision of Hans Rådström.<!-- Following Rådström's 1970 death, Enflo supervised the thesis of another student of Rådström, also on Hilbert's fifth problem.-->]]
+[[Image:Per Enflo.jpg|right|thumb|[[Per Enflo]] ''(pictured)'' wrote his doctoral thesis under the supervision of Hans Rådström.<!-- Following Rådström's 1970 death, Enflo supervised the thesis of another student of Rådström, also on Hilbert's fifth problem.-->]]
 Rådström was interested in [[Hilbert's fifth problem]] on the analyticity of the continuous operation of [[topological group]]s. The solution of this problem by [[Andrew Gleason]] used constructions of [[subset]]s of [[topological vector space]]s,<ref>{{cite book|first=Andrew |last=Gleason|author-link=Andrew Gleason|chapter=One-parameter subgroups and Hilbert's fifth problem|pages=451&ndash;452|volume=2|title=Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, 1950|location=Providence, Rhode Island|publisher=American Mathematical Society|year=1952}}</ref> (rather than simply [[singleton (mathematics)|point]]s), and inspired Rådström's research on [[set-valued analysis]].
-He visited the [[Institute for Advanced Study]] (IAS) in Princeton from 1948 to 1950,<ref>{{cite web|url=http://www.math.ias.edu/people/past-alpha?letter=R|title=Past Members Alphabetical: R|publisher=[[Institute for Advanced Study]]|year=2011|accessdate=29 December 2011}}</ref> where he co-organized a seminar on convexity.<ref>{{cite journal|mr=0064421|last1=Bateman|first1=P. T.|authorlink1=Paul T. Bateman |last2=Rådström|first2=Hans|last3=Hanner|first3=Olaf|author3-link=Olof Hanner|last4=Macbeath|first4=A. M.|authorlink4=A. M. Macbeath|last5=Rogers|first5=C. A.|authorlink5=Claude Ambrose Rogers|last6=Pettis|first6=B. J.|authorlink6=B. J. Pettis|last7=Klee|first7=V. L.|authorlink7=Victor Klee|title=Seminar on convex sets, 1949–1950|publisher=The Institute for Advanced Study|location=Princeton, N. J.}}</ref> 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 theorem (convex hull)|Carathéodory's lemma]].<ref>{{cite journal|title=Generalizations of a theorem of Carathéodory|first=John R.|last=Reay|journal=Mem. Amer. Math. Soc.|volume=54|year=1965|ref=harv|mr=0188891|institution=Mathematics Department, University of Washington|format=Doctoral thesis}}</ref>
+He visited the [[Institute for Advanced Study]] (IAS) in Princeton from 1948 to 1950,<ref>{{cite web|url=https://www.ias.edu/scholars/hans-v-r%C3%A5dstr%C3%B6m|title=IAS Past Member Profile|publisher=[[Institute for Advanced Study]]|year=2011|access-date=16 October 2021}}</ref> where he co-organized a seminar on convexity.<ref>{{cite book|mr=0064421|last1=Bateman|first1=P. T.|author-link1=Paul T. Bateman |last2=Rådström|first2=Hans|last3=Hanner|first3=Olaf|author3-link=Olof Hanner|last4=Macbeath|first4=A. M.|author-link4=A. M. Macbeath|last5=Rogers|first5=C. A.|author-link5=Claude Ambrose Rogers|last6=Pettis|first6=B. J.|author-link6=B. J. Pettis|last7=Klee|first7=V. L.|author-link7=Victor Klee|title=Seminar on convex sets, 1949–1950|publisher=The Institute for Advanced Study|location=Princeton, N. J.}}</ref> 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 theorem (convex hull)|Carathéodory's lemma]].<ref>{{cite journal|title=Generalizations of a theorem of Carathéodory|first=John R.|last=Reay|journal=Mem. Amer. Math. Soc.|volume=54|year=1965|mr=0188891|institution=Mathematics Department, University of Washington|type=Doctoral thesis}}</ref>
-In the 1950s, he obtained important results on [[convex set]]s. He proved the ''Rådström embedding theorem'', which implies<!-- His general conditions are stated on page 58; they also cover the finite-dimensional [[compact space|compact]] convex sets --> that the collection of all [[empty set|nonempty]] compact convex subsets of a [[normed space|normed]] real vector-space (endowed with the [[Hausdorff distance]]) can be [[isometry|isometrically]] embedded as a [[convex cone]] in a normed real vector-space. Under the embedding, the nonempty compact convex sets are mapped to [[singleton (mathematics)|point]]s in the [[range (mathematics)|range]] space. In Rådström's construction, this embedding is additive and positively homogeneous.<ref name="Schneider"/> Rådström's approach used ideas from the theory of topological semi-groups.<ref name="Schmidt">{{cite journal|last=Schmidt|first=Klaus D|journal=Acta Applicandae Mathematicae|volume=5|issue=3|pages=209–237|doi=10.1007/BF00047343|mr=|title=Embedding theorems for classes of convex sets|date=March 1986|ref=harv}}</ref> Later, [[Lars Hörmander]] proved a variant of this theorem for [[locally convex topological vector space]]s using the [[support function]] (of [[convex analysis]]); in Hörmander's approach, the range  of the embedding was the [[Banach space|Banach]] [[Riesz space|lattice]] ''L''<sub>1</sub>, and the embedding was [[order-preserving|isotone]].<ref name="Schneider">
+In the 1950s, he obtained important results on [[convex set]]s. He proved the ''Rådström embedding theorem'', which implies<!-- His general conditions are stated on page 58; they also cover the finite-dimensional [[compact space|compact]] convex sets --> that the collection of all [[empty set|nonempty]] compact convex subsets of a [[normed space|normed]] real vector-space (endowed with the [[Hausdorff distance]]) can be [[isometry|isometrically]] embedded as a [[convex cone]] in a normed real vector-space. Under the embedding, the nonempty compact convex sets are mapped to [[singleton (mathematics)|point]]s in the [[range of a function|range]] space. In Rådström's construction, this embedding is additive and positively homogeneous.<ref name="Schneider"/> Rådström's approach used ideas from the theory of topological semi-groups.<ref name="Schmidt">{{cite journal|last=Schmidt|first=Klaus D|journal=Acta Applicandae Mathematicae|volume=5|issue=3|pages=209–237|doi=10.1007/BF00047343|title=Embedding theorems for classes of convex sets|date=March 1986|s2cid=123172020|url=http://madoc.bib.uni-mannheim.de/51056/}}</ref> Later, [[Lars Hörmander]] proved a variant of this theorem for [[locally convex topological vector space]]s using the [[support function]] (of [[convex analysis]]); in Hörmander's approach, the range  of the embedding was the [[Banach space|Banach]] [[Riesz space|lattice]] ''L''<sub>1</sub>, and the embedding was [[order-preserving|isotone]].<ref name="Schneider">
 {{harvtxt|Schneider|1993|loc=Notes for section 1.8 (pp. 56–61, especially 57–58)}}:
-{{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of mathematics and its applications|volume=44|publisher=Cambridge&nbsp;University Press|location=Cambridge|year=1993|pages=xiv+490 |isbn=0-521-35220-7|mr=1216521|ref=harv}}</ref><ref name="Schmidt"/><ref>{{cite book|mr=1301332|last=Hörmander|first=Lars|title=Notions of convexity|series=Progress in Mathematics|volume=127|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=1994|isbn=0-8176-3799-0}}</ref>
+{{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of mathematics and its applications|volume=44|publisher=Cambridge&nbsp;University Press|location=Cambridge|year=1993|pages=xiv+490|isbn=978-0-521-35220-8|mr=1216521|doi=10.1017/CBO9780511526282|url=https://archive.org/details/convexbodiesbrun0000schn|url-access=registration}}</ref><ref name="Schmidt"/><ref>{{cite book|mr=1301332|last=Hörmander|first=Lars|title=Notions of convexity|series=Progress in Mathematics|volume=127|publisher=Birkhäuser Boston, Inc.|location=Boston, MA|year=1994|isbn=978-0-8176-3799-6}}</ref>
-Rådström characterized the generators of [[topological semigroup|continuous semigroup]]s of sets as [[compact space|compact]] convex sets.<ref>{{cite book|title=Lie groups, convex cones, and semigroups|last1=Hilgert|first1=Joachim|authorlink1=Joachim Hilgert|last2=Hofmann|first2=Karl Heinrich|authorlink2=Karl Heinrich Hofmann|last3=Lawson|first3=Jimmie&nbsp;D.|authorlink3=Jimmie D. Lawson|isbn=978-0-19-853569-0|lccn=89009289|series=Oxford Mathematical Monographs|url=https://books.google.com/books?hl=sv&id=TbyxAAAAIAAJ&q=Radstrom#search_anchor|year=1989|publisher=Oxford University Press|ref=harv}}</ref>
+Rådström characterized the generators of [[topological semigroup|continuous semigroup]]s of sets as [[compact space|compact]] convex sets.<ref>{{cite book|title=Lie groups, convex cones, and semigroups|last1=Hilgert|first1=Joachim|author-link1=Joachim Hilgert|last2=Hofmann|first2=Karl Heinrich|author-link2=Karl Heinrich Hofmann|last3=Lawson|first3=Jimmie&nbsp;D.|author-link3=Jimmie D. Lawson|isbn=978-0-19-853569-0|lccn=89009289|series=Oxford Mathematical Monographs|url=https://books.google.com/books?id=TbyxAAAAIAAJ&q=Radstrom|year=1989|publisher=Oxford University Press}}</ref>
 ==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 space|uniform]] and [[Lipschitz continuity|Lipschitz]] [[category theory|categories]] of [[topological vector space]]s, Enflo's results<ref>{{cite book|last=Enflo|first=Per|author-link=per Enflo|year=1970|title=Investigations on Hilbert’s fifth problem for non locally compact groups|publisher=Stockholm University|format=doctoral thesis}}</ref> concerned [[locally convex topological vector space|spaces with local convexity]], especially [[Banach space]]s.<ref name="GNFA">{{cite book|last1=Lindensrauss|first1=Joram|author1-link=Joram Lindenstrauss|last2=Benyamini|first2=Yoav|title=Geometric nonlinear functional analysis|series=Colloquium publications|volume=48|publisher=American Mathematical Society}}</ref><ref name="LDG">
+Rådström's Ph.D. students included [[Per Enflo]] and [[Martin Ribe]], both of whom wrote Ph.D. theses in [[functional analysis]]. In the [[uniform space|uniform]] and [[Lipschitz continuity|Lipschitz]] [[category theory|categories]] of [[topological vector space]]s, Enflo's results<ref>{{cite book|last=Enflo|first=Per|author-link=per Enflo|year=1970|title=Investigations on Hilbert's fifth problem for non locally compact groups|publisher=Stockholm University|type=doctoral thesis}}</ref> concerned [[locally convex topological vector space|spaces with local convexity]], especially [[Banach space]]s.<ref name="GNFA">{{cite book|last1=Lindensrauss|first1=Joram|author1-link=Joram Lindenstrauss|last2=Benyamini|first2=Yoav|title=Geometric nonlinear functional analysis|series=Colloquium publications|volume=48|publisher=American Mathematical Society}}</ref><ref name="LDG">
-{{cite book|authorlink=Jiří Matoušek (mathematician)|last=Matoušek|first=Jiří|title=Lectures on Discrete Geometry|url=http://kam.mff.cuni.cz/~matousek/dg-nmetr.ps.gz|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|year=2002|isbn=978-0-387-95373-1}}</ref><!--<ref>
+{{cite book|author-link=Jiří Matoušek (mathematician)|last=Matoušek|first=Jiří|title=Lectures on Discrete Geometry|url=http://kam.mff.cuni.cz/~matousek/dg-nmetr.ps.gz|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|year=2002|isbn=978-0-387-95373-1}}</ref><!--<ref>
 * Enflo, Per. (1970) Investigations on Hilbert’s fifth problem for non locally compact groups (Stockholm University). Enflo's thesis contains reprints of exactly five papers:
 ** Enflo, Per; 1969a: Topological groups in which multiplication on one side is differentiable or linear. ''Math. Scand.,'' 24, pp. 195–197.
@@ Line 67: / Line 66: @@
 |issue=2|bibcode = 1970ArM.....8..103E }}
 ** Enflo, Per; 1969b: On a problem of Smirnov. ''Ark. Math''., 8, pp. 107–109.
-** {{cite journal | last1 = Enflo | first1 = Per | year = 1970a | title = Uniform structures and square roots in topological groups '''I''' | url = | journal = Israel J. Math. | volume = 8 | issue = 3| pages = 230–252 | doi = 10.1007/BF02771560 }}
+** {{cite journal | last1 = Enflo | first1 = Per | year = 1970a | title = Uniform structures and square roots in topological groups '''I''' | journal = Israel J. Math. | volume = 8 | issue = 3| pages = 230–252 | doi = 10.1007/BF02771560 }}
-** {{cite journal | last1 = Enflo | first1 = Per | year = 1970b | title = Uniform structures and square roots in topological groups '''II''' | url = | journal = Israel J. Math. | volume = 8 | issue = 3| pages = 253–272 | doi = 10.1007/BF02771561 }}</ref>  Enflo earned an advanced Ph.D. at Stockholm University in 1970, where Rådström was a [[docent]]. In 1969, Rådström was appointed as the Professor of Applied Mathematics at [[Linköping University]].-->
+** {{cite journal | last1 = Enflo | first1 = Per | year = 1970b | title = Uniform structures and square roots in topological groups '''II''' | journal = Israel J. Math. | volume = 8 | issue = 3| pages = 253–272 | doi = 10.1007/BF02771561 }}</ref>  Enflo earned an advanced Ph.D. at Stockholm University in 1970, where Rådström was a [[docent]]. In 1969, Rådström was appointed as the Professor of Applied Mathematics at [[Linköping University]].-->
 In 1970,<ref>{{harvtxt|Kiselman|2010|p=1436}}:
+{{cite journal|title=Inverses and quotients of mappings between ordered sets|journal=Image and Vision Computing|volume=28|issue=10|year=2010|pages=1429–1442|first=Christer O.|last=Kiselman|doi=10.1016/j.imavis.2009.06.014}}</ref> Hans Rådström died of a [[heart attack]].<ref  name="EnfloWords" >{{cite web|title=Personal notes, in my own words|first=Per|last=Enflo|author-link=Per Enflo|date=25 April 2011|access-date=13 December 2011|publisher=perenflo.com|url=http://perenflo.com/sida9.html|archive-url=https://web.archive.org/web/20120426050617/http://perenflo.com/sida9.html|archive-date=26 April 2012|url-status=dead}}</ref> Enflo supervised one of Rådström's Linköping students, Lars-Erik Andersson, from 1970–1971, helping him with his 1972 thesis,<ref name="EnfloWords"/>  ''On connected subgroups of Banach spaces'', on [[Hilbert's fifth problem]] for [[Banach space|complete, normed space]]s. The Swedish [[functional analysis|functional analyst]] [[Edgar Asplund]], then Professor of Mathematics at [[Aarhus University]] in Denmark, assisted Ribe as supervisor of his  1972 thesis,<ref>Acknowledgement in {{cite book|last=Ribe|first=Martin|year=1972|title=On Spaces Which Are Not Supposed to be Locally Convex|type=doctoral thesis|location=Linköping|publisher=Högsk}}</ref> before dying of cancer in 1974.<ref>{{cite journal|mr=2362689|last=Borwein|first=Jonathan M.|author-link=Jonathan Borwein|title=Asplund decompositions of monotone operators|pages=19&ndash;25|journal=ESAIM Proc.|volume=17|year=2007|doi=10.1051/proc:071703|doi-access=free}}</ref> Ribe's results concerned  topological vector spaces without assuming local convexity;<ref name="GNFA"/> Ribe constructed a counter-example to naive extensions of the [[Hahn–Banach theorem]] to topological vector spaces that lack local convexity.<ref>{{cite book
-{{cite journal|title=Inverses and quotients of mappings between ordered sets|journal=Image and Vision Computing|volume=28|year=2010|pages=1429–1442|url=|first=Christer O.|last=Kiselman|ref=harv|format=pdf|doi=10.1016/j.imavis.2009.06.014}}</ref> Hans Rådström died of a [[heart attack]].<ref  name="EnfloWords" >
-{{cite web|title=Personal notes, in my own words|first=Per|last=Enflo|authorlink=Per Enflo|date=25 April 2011|accessdate=13 December 2011|ref=harv|publisher=perenflo.com|url=http://perenflo.com/sida9.html}}
-</ref> Enflo supervised one of Rådström's Linköping students, Lars-Erik Andersson, from 1970–1971, helping him with his 1972 thesis,<ref name="EnfloWords"/>  ''On connected subgroups of Banach spaces'', on [[Hilbert's fifth problem]] for [[Banach space|complete, normed space]]s. The Swedish [[functional analysis|functional analyst]] [[Edgar Asplund]], then Professor of Mathematics at [[Aarhus University]] in Denmark, assisted Ribe as supervisor of his  1972 thesis,<ref>Acknowledgement in {{cite book|last=Ribe|first=Martin|year=1972|title=On Spaces Which Are Not Supposed to be Locally Convex|format=doctoral thesis|location=Linköping|publisher=Högsk}}</ref> before dying of cancer in 1974.<ref>{{cite journal|mr=2362689|last=Borwein|first=Jonathan M.|authorlink=Jonathan Borwein|title=Asplund decompositions of monotone operators|pages=19&ndash;25|journal=ESAIM Proc.|volume=17|year=2007|url=http://web.cs.dal.ca/~jborwein/decomps-paper.pdf|format=pdf|accessdate=13 December 2011|doi=10.1051/proc:071703|deadurl=yes|archiveurl=https://web.archive.org/web/20120415101530/http://web.cs.dal.ca/~jborwein/decomps-paper.pdf|archivedate=15 April 2012|df=}}</ref> Ribe's results concerned  topological vector spaces without assuming local convexity;<ref name="GNFA"/> Ribe constructed a counter-example to naive extensions of the [[Hahn–Banach theorem]] to topological vector spaces that lack local convexity.<ref>{{cite book
 |last1=Kalton|first1=Nigel J.|author1-link=Nigel Kalton
 |last2=Peck|first2=N. Tenney
@@ Line 79: / Line 76: @@
 | title = An F-space sampler
 | series = London Mathematical Society Lecture Note Series|volume=89
-| publisher = Cambridge University Press| publication-place = Cambridge
+| publisher = Cambridge University Press| location = Cambridge
-| year = 1984| pages = xii+240| isbn = 0-521-27585-7|mr=808777|ref=harv}}</ref>
+| year = 1984| pages = xii+240| isbn = 978-0-521-27585-9|mr=808777|doi=10.1017/CBO9780511662447}}</ref>
 ==References==
@@ Line 96: / Line 93: @@
 [[Category:Functional analysts]]
 [[Category:Mathematical analysts]]
-[[Category:20th-century mathematicians]]
+[[Category:20th-century Swedish mathematicians]]
-[[Category:Swedish mathematicians]]