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:^[1]

Let X be a paracompact space and Y a Banach space.

Let ${\displaystyle F\colon X\to Y}$ be a lower hemicontinuous set-valued function with nonempty convex closed values.

Then there exists a continuous selection ${\displaystyle f\colon X\to Y}$ of F.

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: ${\displaystyle F(x)=[1-x/2,~1-x/4]}$, 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: ${\displaystyle f(x)=1-x/2}$ or ${\displaystyle f(x)=1-3x/8}$.

A function that does not satisfy lower hemicontinuity

The function

${\displaystyle F(x)={\begin{cases}3/4&0\leq x<0.5\\\left[0,1\right]&x=0.5\\1/4&0.5<x\leq 1\end{cases}}}$

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.^[2]

Applications

Michael selection theorem can be applied to show that the differential inclusion

${\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),\quad x(t_{0})=x_{0}}$

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 ${\displaystyle F}$ is said to be almost lower hemicontinuous if at each ${\displaystyle x\in X}$, all neighborhoods ${\displaystyle V}$ of ${\displaystyle 0}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ such that ${\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .}$

Precisely, Deutsch–Kenderov theorem states that if ${\displaystyle X}$ is paracompact, ${\displaystyle Y}$ a normed vector space and ${\displaystyle F(x)}$ is nonempty convex for each ${\displaystyle x\in X}$, then ${\displaystyle F}$ is almost lower hemicontinuous if and only if ${\displaystyle F}$ has continuous approximate selections, that is, for each neighborhood ${\displaystyle V}$ of ${\displaystyle 0}$ in ${\displaystyle Y}$ there is a continuous function ${\displaystyle f\colon X\mapsto Y}$ such that for each ${\displaystyle x\in X}$, ${\displaystyle f(x)\in F(X)+V}$.^[3]

In a note Xu proved that Deutsch–Kenderov theorem is also valid if ${\displaystyle Y}$ is a locally convex topological vector space.^[4]

See also

• Zero-dimensional Michael selection theorem
• Selection theorem
• Maximum theorem

References

Further reading

• Repovš, Dušan; Semenov, Pavel V. (2014). "Continuous Selections of Multivalued Mappings". In Hart, K. P.; van Mill, J.; Simon, P. (eds.). Recent Progress in General Topology. Vol. III. Berlin: Springer. pp. 711–749. arXiv:1401.2257. Bibcode:2014arXiv1401.2257R. ISBN 978-94-6239-023-2.
• Aubin, Jean-Pierre; Cellina, Arrigo (1984). Differential Inclusions, Set-Valued Maps And Viability Theory. Grundl. der Math. Wiss. Vol. 264. Berlin: Springer-Verlag. ISBN 3-540-13105-1.
• Aubin, Jean-Pierre; Frankowska, H. (1990). Set-Valued Analysis. Basel: Birkhäuser. ISBN 3-7643-3478-9.
• Deimling, Klaus (1992). Multivalued Differential Equations. Walter de Gruyter. ISBN 3-11-013212-5.
• Repovš, Dušan; Semenov, Pavel V. (1998). Continuous Selections of Multivalued Mappings. Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-5277-7.
• Repovš, Dušan; Semenov, Pavel V. (2008). "Ernest Michael and Theory of Continuous Selections". Topology and its Applications. 155 (8): 755–763. arXiv:0803.4473. doi:10.1016/j.topol.2006.06.011.
• Aliprantis, Charalambos D.; Border, Kim C. (2007). Infinite Dimensional Analysis : Hitchhiker's Guide (3rd ed.). Springer. ISBN 978-3-540-32696-0.
• Hu, S.; Papageorgiou, N. Handbook of Multivalued Analysis. Vol. I. Kluwer. ISBN 0-7923-4682-3.