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

1. For every closed convex balanced neighborhood Template:Mvar of the origin in Template:Mvar, there exists a neighborhood Template:Mvar of Template:Math in Template:Mvar such that for all real Template:Math, Template:Mvar can be covered by finitely many translates of Template:Math.
2. Every bounded subset of Template:Mvar is totally bounded and for every closed convex balanced neighborhood Template:Mvar of the origin in Template:Mvar, there exists a neighborhood Template:Mvar of Template:Math in Template:Mvar such that for all real Template:Math, there exists a bounded subset Template:Mvar of Template:Mvar such that Template:Math.

Properties

Every quasi-complete Schwartz space is a semi-Montel space. Every Fréchet Schwartz space is a Montel space.Template:Sfn

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.Template:Sfn

There exist Fréchet spaces that are not Schwartz spaces and there exist Schwartz spaces that are not Montel spaces.Template:Sfn

See also

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Bibliography

• Template:Cite journal
• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• Template:Robertson Topological Vector Spaces
• Template:Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces
• Template:Jarchow Locally Convex Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels