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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:

Contents

Let E be a Banach space, X a paracompact space and F : XE a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f : XE 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

d x d t ( t ) F ( t , x ( t ) ) , x ( t 0 ) = x 0

has a C1 solution when F is lower semi-continuous and F(tx) is a nonempty closed and convex set for all (tx). 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 xX, all neighborhoods V of 0 there exists a neighborhood U of x such that u U { F ( u ) + V } . Precisely, Deutsch and Kenderov theorem states that if X is paracompact, E a normed vector space and F(x) is nonempty convex for each xX, 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:XE such that for each xX, 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.

References

Michael selection theorem Wikipedia


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