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.

Definition

For a locally convex space X with continuous dual $X^{\prime }$, 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

• Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
• Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 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. {{cite book}}: Invalid |ref=harv (help)
• Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1. {{cite book}}: Invalid |ref=harv (help)
• Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. Vol. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. {{cite book}}: Invalid |ref=harv (help)
• Schaefer, Helmut H. (1971). Topological vector spaces. GTM. Vol. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)
• Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. {{cite book}}: Invalid |ref=harv (help)