In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, a branch of mathematics, Michael selection theorem is a
selection theorem In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the ...
named after
Ernest Michael. In its most popular form, it states the following:
: Let ''X'' be a
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
space and ''Y'' a
Banach space.
:Let
be a
lower hemicontinuous set-valued function
A set-valued function (or correspondence) is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including optimizatio ...
with nonempty
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
closed values.
:Then there exists a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
selection
Selection may refer to:
Science
* Selection (biology), also called natural selection, selection in evolution
** Sex selection, in genetics
** Mate selection, in mating
** Sexual selection in humans, in human sexuality
** Human mating strateg ...
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
Selection may refer to:
Science
* Selection (biology), also called natural selection, selection in evolution
** Sex selection, in genetics
** Mate selection, in mating
** Sexual selection in humans, in human sexuality
** Human mating strateg ...
, then ''X'' is paracompact. This provides another characterization for
paracompactness
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...
.
Examples
A function that satisfies all requirements
The function:
, shown by the grey area in the figure at the right, is a set-valued function from the real interval
,1to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example:
or
.
A function that does not satisfy lower hemicontinuity
The function
is a set-valued function from the real interval
,1to 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
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form
:\frac(t)\in F(t,x(t)),
where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point ...
:
has a ''C''
1 solution when ''F'' is
lower semi-continuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ...
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
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees t ...
.
Generalizations
A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to an equivalence relating approximate selections to almost
lower hemicontinuity, where
is said to be almost lower hemicontinuous if at each
, all neighborhoods
of
there exists a neighborhood
of
such that
Precisely, Deutsch–Kenderov theorem states that if
is paracompact,
a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
and
is nonempty convex for each
, then
is almost
lower hemicontinuous if and only if
has continuous approximate selections, that is, for each neighborhood
of
in
there is a continuous function
such that for each
,
.
In a note Xu proved that Deutsch–Kenderov theorem is also valid if
is a locally convex
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
.
See also
*
Zero-dimensional Michael selection theorem
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical i ...
*
Selection theorem In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the ...
References
Further reading
*
*
*
*
*
*
*
*
{{Functional analysis
Theory of continuous functions
Properties of topological spaces
Theorems in functional analysis
Compactness theorems