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 : 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.

Applications

Michael selection theorem can be applied to show that the differential inclusion

$\frac{dx}{dt}(t)\in F(t,x(t)), \quad x(t_0)=x_0$

has a C¹ solution when F is lower semi-continuous and F(t, x) is a nonempty closed and convex set for all (t, x). When F is single valued, this is the classic Peano existence theorem.

See also

• Zero-dimensional Michael selection theorem
• List of selection theorems

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, MR 0077107
• Dušan Repovš; Pavel V. Semenov (2014). "Continuous Selections of Multivalued Mappings". In Hart, K. P.; van Mill, J.; Simon, P. Recent progress in general topology III. Berlin: Springer. pp. 711–749. ISBN 978-94-6239-023-2.
• 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