In functional analysis, a branch of mathematics, the most popular version of the Michael selection theorem, named after Ernest Michael, states the following:
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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.Applications
Michael selection theorem can be applied to show that the differential inclusion
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.
Generalizations
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
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.