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 ${\displaystyle X^{\prime }}$, X is called a Schwartz space if it satisfies any of the following equivalent conditions:^[1]

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.^[2]

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.^[2]

There exist Fréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces.^[2]

See also

• Auxiliary normed space
• Montel space – Barrelled space where closed and bounded subsets are compact