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:
Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values, admits a continuous selection, then X is paracompact. This provides another characterization for paracompactness.
Examples
A function that satisfies all requirements The function: F ( x ) = [ 1 − x / 2 , 1 − x / 4 ] {\displaystyle F(x)=[1-x/2,~1-x/4]} , shown by the grey area in the figure at the right, is a set-valued function from the real interval [0,1] to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: f ( x ) = 1 − x / 2 {\displaystyle f(x)=1-x/2} or f ( x ) = 1 − 3 x / 8 {\displaystyle f(x)=1-3x/8} .
A function that does not satisfy lower hemicontinuity The function
F ( x ) = { 3 / 4 0 ≤ x < 0.5 [ 0 , 1 ] x = 0.5 1 / 4 0.5 < x ≤ 1 {\displaystyle F(x)={\begin{cases}3/4&0\leq x<0.5\\\left[0,1\right]&x=0.5\\1/4&0.5<x\leq 1\end{cases}}}
is a set-valued function from the real interval [0,1] 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
d x d t ( t ) ∈ F ( t , x ( t ) ) , x ( t 0 ) = x 0 {\displaystyle {\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.
Generalizations A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to an equivalence relating approximate selections to almost lower hemicontinuity, where F {\displaystyle F} is said to be almost lower hemicontinuous if at each x ∈ X {\displaystyle x\in X} , all neighborhoods V {\displaystyle V} of 0 {\displaystyle 0} there exists a neighborhood U {\displaystyle U} of x {\displaystyle x} such that ∩ u ∈ U { F ( u ) + V } ≠ ∅ . {\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .}
Precisely, Deutsch–Kenderov theorem states that if X {\displaystyle X} is paracompact, Y {\displaystyle Y} a normed vector space and F ( x ) {\displaystyle F(x)} is nonempty convex for each x ∈ X {\displaystyle x\in X} , then F {\displaystyle F} is almost lower hemicontinuous if and only if F {\displaystyle F} has continuous approximate selections, that is, for each neighborhood V {\displaystyle V} of 0 {\displaystyle 0} in Y {\displaystyle Y} there is a continuous function f : X ↦ Y {\displaystyle f\colon X\mapsto Y} such that for each x ∈ X {\displaystyle x\in X} , f ( x ) ∈ F ( X ) + V {\displaystyle f(x)\in F(X)+V} . In a note Xu proved that Deutsch–Kenderov theorem is also valid if Y {\displaystyle Y} is a locally convex topological vector space.
See also Zero-dimensional Michael selection theorem Selection theorem Maximum theorem
References
… excerpt ends here. Continue reading the full article.
