Michael selection theorem
From Wikipedia, the free encyclopedia
- Let E be a Banach space, X a paracompact space and φ : X → E a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f : X → E 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
- Aumann measurable selection theorem
- Bressan-Colombo directionally continuous selection theorem
- Castaing representation theorem
- Fryszkowski decomposable map selection
- Helly's selection theorem
- Kuratowski, Ryll-Nardzewski measurable selection theorem
- Michael, Ernest (1956), "Continuous selections. I", Annals of Mathematics. Second Series (Annals of Mathematics) 63 (2): 361–382, doi:10.2307/1969615, JSTOR 1969615, MR 0077107
- 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
- D.Repovs and P. V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht 1998.
- D.Repovs and P.V.Semenov, Ernest Michael and theory of continuous selections, Topol. Appl. 155:8 (2008), 755-763.
- Aliprantis, Kim C. Border Infinite dimensional analysis. Hitchhiker's guide Springer
- S.Hu, N.Papageorgiou Handbook of multivalued analysis. Vol. I Kluwer