Schwartz topological vector space

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 $X$ with continuous dual $$X^{\prime}$$, $X$ is called a Schwartz space if it satisfies any of the following equivalent conditions:


 * 1) For every closed convex balanced neighborhood $U$ of the origin in $X$, there exists a neighborhood $V$ of $0$ in $X$ such that for all real $r > 0$, $V$ can be covered by finitely many translates of $rU$.
 * 2) Every bounded subset of $X$ is totally bounded and for every closed convex balanced neighborhood $U$ of the origin in $X$, there exists a neighborhood $V$ of $0$ in $X$ such that for all real $r > 0$, there exists a bounded subset $B$ of $X$ such that $V ⊆ B + rU$.

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.