Michael selection theorem


In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following:

Examples

A function that satisfies all requirements

The function:, shown by the grey area in the figure at the right, is a multi-valued function from the real interval to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: or.

A function that does not satisfy lower hemicontinuity

The function
is a multi-valued function from the real interval to itself. It has nonempty convex closed values. However, it is not lower hemicontinuous at 0.5. Indeed, Michael's theorem does not apply and the function does not have a continuous selection: any selection at 0.5 is necessarily discontinuous.

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 is a nonempty closed and convex set for all. 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 an equivalence relating approximate selections to almost lower hemicontinuity, where is said to be almost lower hemicontinuous if at each, all neighborhoods of there exists a neighborhood of such that
Precisely, Deutsch–Kenderov theorem states that if is paracompact, a normed vector space and is nonempty convex for each, then is almost lower hemicontinuous if and only if has continuous approximate selections, that is, for each neighborhood of in there is a continuous function such that for each,.
In a note Xu proved that Deutsch–Kenderov theorem is also valid if is a locally convex topological vector space.