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 F : X → Y {\displaystyle F\colon X\to Y} be a lower hemicontinuous set-valued function with nonempty convex closed values. Then there exists a continuous selection f : X → Y {\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: 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.2
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} .3
In a note Xu proved that Deutsch–Kenderov theorem is also valid if Y {\displaystyle Y} is a locally convex topological vector space.4
See also
- Zero-dimensional Michael selection theorem
- Selection theorem
- Maximum theorem
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.
References
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. /wiki/Ernest_Michael ↩
"proof verification - Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem". Mathematics Stack Exchange. Retrieved 2019-10-29. https://math.stackexchange.com/q/3377063 ↩
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. /wiki/Doi_(identifier) ↩
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. https://doi.org/10.1006%2Fjath.2001.3622 ↩