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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:

Michael Selection Theorem — Let X be a paracompact space and Y be a separable 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: ${\displaystyle F(x)=}$, shown by the grey area in the figure at the right, is a set-valued function from the real interval 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

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

is a set-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

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

has a C 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 ${\displaystyle x\in X}$, ${\displaystyle f(x)\in F(X)+V}$.

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

See also

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

References

1. Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics. Second Series. 63 (2): 361–382. doi:10.2307/1969615. hdl:10338.dmlcz/119700. JSTOR 1969615. MR 0077107.
2. "proof verification - Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem". Mathematics Stack Exchange. Retrieved 2019-10-29.
3. Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections". SIAM Journal on Mathematical Analysis. 14 (1): 185–194. doi:10.1137/0514015.
4. Xu, Yuguang (December 2001). "A Note on a Continuous Approximate Selection Theorem". Journal of Approximation Theory. 113 (2): 324–325. doi:10.1006/jath.2001.3622.

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.

• Theory of continuous functions
• Properties of topological spaces
• Theorems in functional analysis
• Compactness theorems