Michael selection theorem

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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:[1]

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} {\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} {\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]} {\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} {\displaystyle f(x)=1-x/2} or f ( x ) = 1 − 3 x / 8 {\displaystyle f(x)=1-3x/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}}} {\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}} {\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} {\displaystyle F} is said to be almost lower hemicontinuous if at each x ∈ X {\displaystyle x\in X} {\displaystyle x\in X}, all neighborhoods V {\displaystyle V} {\displaystyle V} of 0 {\displaystyle 0} {\displaystyle 0} there exists a neighborhood U {\displaystyle U} {\displaystyle U} of x {\displaystyle x} {\displaystyle x} such that ∩ u ∈ U { F ( u ) + V } ≠ ∅ . {\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .} {\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .}

Precisely, Deutsch–Kenderov theorem states that if X {\displaystyle X} {\displaystyle X} is paracompact, Y {\displaystyle Y} {\displaystyle Y} a normed vector space and F ( x ) {\displaystyle F(x)} {\displaystyle F(x)} is nonempty convex for each x ∈ X {\displaystyle x\in X} {\displaystyle x\in X}, then F {\displaystyle F} {\displaystyle F} is almost lower hemicontinuous if and only if F {\displaystyle F} {\displaystyle F} has continuous approximate selections, that is, for each neighborhood V {\displaystyle V} {\displaystyle V} of 0 {\displaystyle 0} {\displaystyle 0} in Y {\displaystyle Y} {\displaystyle Y} there is a continuous function f : X ↦ Y {\displaystyle f\colon X\mapsto Y} {\displaystyle f\colon X\mapsto Y} such that for each x ∈ X {\displaystyle x\in X} {\displaystyle x\in X}, f ( x ) ∈ F ( X ) + V {\displaystyle f(x)\in 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} {\displaystyle Y} is a locally convex topological vector space.[4]

See also

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