Michael selection theorem
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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 : X → E a lower semicontinuous 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.
[edit] 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
[edit] References
- Michael, Ernest (1956), "Continuous selections. I", Annals of Mathematics. Second Series 63: 361–382, doi:, MR0077107
- D.Repovs, P.V.Semenov,Ernest Michael and theory of continuous selections" arXiv:0803.4473v1
- Jean-Paul 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
