Michael selection theorem
- Let E be a Banach space, X a paracompact space and F : X → E a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f : X → E of F.
- 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.
Michael selection theorem can be applied to show that the differential inclusion
A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to a equivalence relating approximate selections to almost lower hemicontinuity, where F is said to be almost lower hemicontinuous if at each x∈X, all neighborhoods V of 0 there exists a neighborhood U of x such that Precisely, Deutsch and Kenderov theorem states that if X is paracompact, E a normed vector space and F(x) is nonempty convex for each x∈X, then F is almost lower hemicontinuous if and only if F has continuous approximate selections, that is, for each neighborhood V of 0 in E there is a continuous function f:X → E such that for each x ∈ X, f(x) ∈ F(X) + V.
In a note of Y. Xu it is proved that Deutsch and Kenderov theorem is also valid if E is locally convex topological vector space.
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