DF-space

In the field of functional analysis, DF-spaces, also written (DF)-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the theory of topological tensor products.[1]

DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in (Grothendieck 1954) harv error: no target: CITEREFGrothendieck1954 (help). 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).[2]

Definition

A locally convex topological vector space (TVS) X is a DF-space, also written (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 ${\displaystyle B_{i}}$[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 B ⊆ X, B ∩ 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]
• Every infinite-dimensional Montel DF-space is a sequential space but not a Fréchet–Urysohn space.
• Suppose X is either a DF-space or an LM-space. If X is a sequential space then it is either metrizable or else a Montel space DF-space.
• Every quasi-complete DF-space is complete.[5]
• If X is a complete nuclear DF-space then X is a Montel space.[6]

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.[7]
  • Every Banach space is a DF-space.[1]
  • Every infrabarreled space possessing a fundamental sequence of bounded sets is a DF-space.
• Every Hausdorff quotient of a DF-space is a DF-space.[4]
• The completion of a DF-space is a DF-space.[4]
• The locally convex sum of a sequence of DF-spaces is a DF-space.[4]
• An inductive limit of a sequence of DF-spaces is a DF-space.[4]
• Suppose that X and Y are DF-spaces. Then the projective tensor product, as well as its completion, of these spaces is a DF-space.[6]

However,

• An infinite product of non-trivial DF-spaces (i.e. all factors have non-0 dimension) is not a DF-space.[4]
• A closed vector subspace of a DF-space is not necessarily a DF-space.[4]
• There exist complete DF-spaces that are not TVS-isomorphic to the strong dual of a metrizable locally convex TVS.[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.[8]

See also

• Barreled space
• Countably quasi-barrelled space
• F-space – Topological vector space with a complete translation-invariant metric
• LB-space
• LF-space
• Nuclear space – Type of topological vector space
• Projective tensor product

Citations

1. Schaefer & Wolff 1999, pp. 154-155.
2. Schaefer & Wolff 1999, pp. 152,154.
3. Schaefer & Wolff 1999, p. 25.
4. Schaefer & Wolff 1999, pp. 196-197.
5. Schaefer & Wolff 1999, pp. 190-202.
6. Schaefer & Wolff 1999, pp. 199-202.
7. Khaleelulla 1982, p. 33.
8. Khaleelulla 1982, pp. 103-110.

Bibliography

• Grothendieck, Alexander (1954). "Sur les espaces (F) et (DF)". Summa Brasil. Math. (in French). 3: 57–123. MR 0075542.
• Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-05644-9. OCLC 539541.
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin,New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541.
• 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.
• Wong, Yau-Chuen (1979). Schwartz Spaces, Nuclear Spaces, and Tensor Products. Lecture Notes in Mathematics. 726. Berlin New York: Springer-Verlag. ISBN 978-3-540-09513-2. OCLC 5126158.

External links

• DF-space at ncatlab