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 [math]\displaystyle{ X^{\prime} }[/math], 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.

See also

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

Bibliography

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques" (in French). Annales de l'Institut Fourier 2: 5–16 (1951). doi:10.5802/aif.16. http://www.numdam.org/item?id=AIF_1950__2__5_0. 
• Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. 2. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. http://www.numdam.org/item?id=AIF_1950__2__5_0. 
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. 
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. 
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. 
• Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. 
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.