Michael selection theorem

Michael selection theorem

In functional analysis, a branch of mathematics, the most popular version of the Michael selection theorem, named after Ernest Michael, states the following:

Let E be a Banach space, X a paracompact space and φ : XE a lower semicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f : XE of φ.
Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values admits continuous selection, then X is paracompact. This provides another characterization for paracompactness.

Other selection theorems

  • Zero-dimensional Michael selection theorem
  • Castaing representation theorem
  • Bressan-Colombo directionally continuous selection theorem
  • Fryszkowski decomposable map selection
  • Aumann measurable selection theorem
  • Kuratowski, Ryll-Nardzewski measurable selection theorem

References

  • Michael, Ernest (1956), "Continuous selections. I", Annals of Mathematics. Second Series (Annals of Mathematics) 63 (2): 361–382, doi:10.2307/1969615, JSTOR 1969615, MR0077107 
  • D.Repovs, P.V.Semenov,Ernest Michael and theory of continuous selections" arXiv:0803.4473v1
  • Jean-Pierre Aubin, Arrigo Cellina Differential Inclusions, Set-Valued Maps And Viability Theory, Grundl. der Math. Wiss., vol. 264, Springer - Verlag, Berlin, 1984
  • J.-P. Aubin and H. Frankowska Set-Valued Analysis, Birkh¨auser, Basel, 1990
  • Klaus Deimling Multivalued Differential Equations, Walter de Gruyter, 1992
  • Aliprantis, Kim C. Border Infinite dimensional analysis. Hitchhiker's guide Springer
  • S.Hu, N.Papageorgiou Handbook of multivalued analysis. Vol. I Kluwer

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