Michael selection theorem
Encyclopedia
In functional analysis
, a branch of mathematics, the most popular version of the Michael selection theorem, named after Ernest Michael
, states the following:
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a branch of mathematics, the most popular version of the Michael selection theorem, named after Ernest Michael
Ernest Michael
Ernest A. Michael is an American mathematician who is best known for the Michael selection theorem, which he proved in . He received his Ph.D. from the University of Chicago in 1951. He wrote his dissertation on "Locally m-convex algebras" under the supervision of Irving Segal. He had 5...
, states the following:
- Let E be a Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, X a paracompact space and φ : X → E a lower semicontinuousHemicontinuityIn mathematics, the notion of the continuity of functions is not immediately extendible to multi-valued mappings or correspondences. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extention...
multivalued mapMultivalued functionIn mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
with nonempty convexConvex setIn Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
closedClosed setIn geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
values. Then there exists a continuous selection f : X → E of φ.
- ConverselyConverse (logic)In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P. For the categorical proposition All S is P, the converse is All P is S. In neither case does the converse necessarily follow from...
, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values admits continuous selection, then X is paracompact. This provides another characterization for paracompactnessParacompact spaceIn mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
.
Other selection theorems
- Zero-dimensional Michael selection theorem
- Castaing representation theorem
- Bressan-Colombo directionally continuous selection theorem
- Fryszkowski decomposable map selection
- AumannRobert AumannRobert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
measurable selection theorem - Kuratowski, Ryll-Nardzewski measurable selection theorem