• Source: 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:

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

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






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.

Kata Kunci Pencarian:

• George R. Price
• Bilangan prima
• Teorema Bohr–Mollerup
• James Clerk Maxwell
• Michael selection theorem
• Selection theorem
• Michael's theorem
• Rellich–Kondrachov theorem
• Hemicontinuity
• Set-valued function
• Paracompact space
• Maximum theorem
• Ernest Michael
• Inverse function theorem