In functional analysis and related areas of mathematics, Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets. These spaces were introduced by Alexander Grothendieck.
Definition
A Hausdorff locally convex space with continuous dual , is called a Schwartz space if it satisfies any of the following equivalent conditions:
- For every closed convex balanced neighborhood of the origin in , there exists a neighborhood of in such that for all real , can be covered by finitely many translates of .
- Every bounded subset of is totally bounded and for every closed convex balanced neighborhood of the origin in , there exists a neighborhood of in such that for all real , there exists a bounded subset of such that .
Properties
Every quasi-complete Schwartz space is a semi-Montel space. Every Fréchet Schwartz space is a Montel space.
The strong dual space of a complete Schwartz space is an ultrabornological space.
Examples and sufficient conditions
- Vector subspace of Schwartz spaces are Schwartz spaces.
- The quotient of a Schwartz space by a closed vector subspace is again a Schwartz space.
- The Cartesian product of any family of Schwartz spaces is again a Schwartz space.
- The weak topology induced on a vector space by a family of linear maps valued in Schwartz spaces is a Schwartz space if the weak topology is Hausdorff.
- The locally convex strict inductive limit of any countable sequence of Schwartz spaces (with each space TVS-embedded in the next space) is again a Schwartz space.
Counter-examples
Every infinite-dimensional normed space is not a Schwartz space.
There exist Fréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces.
See also
References
Bibliography