DF-space

In the field of functional analysis, a locally convex topological vector space (TVS) X is a DF-space if[1]

1. X is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of ${\displaystyle X^{\prime }}$ is equicontinuous), and
2. X possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets ${\displaystyle B_{1},B_{2},\ldots }$ such that every bounded subset of X is contained in some ${\displaystyle B_{i}}$[2]).

DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in (Grothendieck 1954). They also play a considerable part in the theory of topological tensor products.[1] Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If X is a metrizable locally convex space and ${\displaystyle V_{1},V_{2},\ldots }$ is a sequence of convex 0-neighborhoods in ${\displaystyle X_{b}^{\prime }}$ such that ${\displaystyle V:=\cap _{i}V_{i}}$ absorbs every strongly bounded set, then V is a 0-neighborhood in ${\displaystyle X_{b}^{\prime }}$ (where ${\displaystyle X_{b}^{\prime }}$ is the continuous dual space of X endowed with the strong dual topology).[3]

Properties

• Let X be a DF-space and let V be a convex balanced subset of X. Then V is a neighborhood of the origin if and only if for every convex, balanced, bounded subset ${\displaystyle B\subseteq X}$, ${\displaystyle B\cap V}$ is a 0-neighborhood in B.[1]
  • Thus, a linear map from a DF-space into a locally convex space is continuous if its restriction to each bounded subset of the domain is continuous.[1]
• The strong dual of a DF-space is a Fréchet space.[4]

Sufficient conditions

• The strong dual of a metrizable locally convex space is a DF-space (but not conversely, in general).[1] Hence:
  • Every normed space is a DF-space.[5]
  • Every Banach space is a DF-space.[1]
  • Every infrabarreled space possessing a fundamental sequence of bounded sets is a DF-space.
• Every separated quotient of a DF-space is a DF-space.[4]

Examples

There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space.[4] There exist DF-spaces spaces having closed vector subspaces that are not DF-spaces.[6]

See also

• Barreled space
• Countably quasi-barrelled space
• Nuclear space
• Projective tensor product

References

1. Schaefer 1999, pp. 154-155. sfn error: no target: CITEREFSchaefer1999 (help)
2. Schaefer 1999, p. 25. sfn error: no target: CITEREFSchaefer1999 (help)
3. Schaefer 1999, pp. 152,154. sfn error: no target: CITEREFSchaefer1999 (help)
4. Schaefer 1999, pp. 196-197. sfn error: no target: CITEREFSchaefer1999 (help)
5. Khaleelulla 1982, p. 33. sfn error: no target: CITEREFKhaleelulla1982 (help)
6. Khaleelulla 1982, pp. 103-110. sfn error: no target: CITEREFKhaleelulla1982 (help)

• Grothendieck, Alexandre (1954). "Sue les espaces (F) et (DF)". Summa Brasil. Math. 3. Cite journal requires |journal= (help)CS1 maint: ref=harv (link)
• Grothendieck, Alexandre (1955). "Produits tensoriels topologiques et espaces nucléaires". Mem. Am. Math. Soc. 16. Cite journal requires |journal= (help)CS1 maint: ref=harv (link)
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Nlend, H (1981). Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality. Amsterdam New York New York, N.Y: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland. ISBN 0-444-86207-2. OCLC 7553061.CS1 maint: ref=harv (link)
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin,New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.CS1 maint: ref=harv (link)
• Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.CS1 maint: ref=harv (link)

External links

• DF-space at ncatlab