Schwartz TVS

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.

Definition

For a 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:

• 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 scalars r > 0, V can be covered by finitely many translates of rU.
• 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 scalars r > 0, there exists a bounded subset B of X such that V ⊆ B + rU.

Properties

Every Fréchet Schwartz space is a Montel space.

Examples

Counter-examples

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

Every infinite-dimensional normed space is not a Schwartz space.

See also

• Montel space

References

• "Sur certains espaces vectoriels topologiques" (in French). Annales de l'Institut Fourier 2: 5–16 (1951). 1950. http://www.numdam.org/item?id=AIF_1950__2__5_0. 
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. pp. 65–75. 
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665. 
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1. 
• 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.