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 ). Grothendieck was led to introduce these spaces by the following property of strong duals of metrizable spaces: If ${\displaystyle X}$ 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 ${\displaystyle V}$ is a 0-neighborhood in ${\displaystyle X_{b}^{\prime }}$ (where ${\displaystyle X_{b}^{\prime }}$ is the continuous dual space of ${\displaystyle X}$ endowed with the strong dual topology). ^[2]

Definition

A locally convex topological vector space (TVS) ${\displaystyle X}$ is a DF-space, also written (DF)-space, if ^[1]

1. ${\displaystyle X}$ is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of ${\displaystyle X^{\prime }}$ is equicontinuous), and
2. ${\displaystyle X}$ 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 ${\displaystyle X}$ is contained in some ${\displaystyle B_{i}}$ ^[3] ).

Properties

• Let ${\displaystyle X}$ be a DF-space and let ${\displaystyle V}$ be a convex balanced subset of ${\displaystyle X.}$ Then ${\displaystyle V}$ 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 neighborhood of the origin in ${\displaystyle B.}$ ^[1] Consequently, 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 space 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 ${\displaystyle X}$ is either a DF-space or an LM-space. If ${\displaystyle X}$ 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 ${\displaystyle X}$ is a complete nuclear DF-space then ${\displaystyle X}$ is a Montel space. ^[6]

Sufficient conditions

The strong dual space ${\displaystyle X_{b}^{\prime }}$ of a Fréchet space ${\displaystyle X}$ is a DF-space. ^[7]

• The strong dual of a metrizable locally convex space is a DF-space ^[8] but the convers is in general not true ^[8] (the converse being the statement that every DF-space is the strong dual of some metrizable locally convex space). From this it follows:
  • Every normed space is a DF-space. ^[9]
  • 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. ^[10]
• The completion of a DF-space is a DF-space. ^[10]
• The locally convex sum of a sequence of DF-spaces is a DF-space. ^[10]
• An inductive limit of a sequence of DF-spaces is a DF-space. ^[10]
• Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ 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. ^[10]
• A closed vector subspace of a DF-space is not necessarily a DF-space. ^[10]
• There exist complete DF-spaces that are not TVS-isomorphic to the strong dual of a metrizable locally convex TVS. ^[10]

Examples

There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space. ^[10] There exist DF-spaces having closed vector subspaces that are not DF-spaces. ^[11]

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  – A generalization of finite dimensional Euclidean spaces different from Hilbert spaces
• 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, p. 196.
5. Schaefer & Wolff 1999, pp. 190–202.
6. Schaefer & Wolff 1999, pp. 199–202.
7. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
8. Schaefer & Wolff 1999, p. 154.
9. Khaleelulla 1982, p. 33.
10. Schaefer & Wolff 1999, pp. 196–197.
11. 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. Vol. 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. Vol. 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. Vol. 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. Vol. 726. Berlin New York: Springer-Verlag. ISBN   978-3-540-09513-2. OCLC   5126158.

External links

• DF-space at ncatlab