DF-space
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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 is a metrizable locally convex space and is a sequence of convex 0-neighborhoods in such that absorbs every strongly bounded set, then is a 0-neighborhood in (where is the continuous dual space of endowed with the strong dual topology).[2]
Definition[]
A locally convex topological vector space (TVS) is a DF-space, also written (DF)-space, if[1]
- is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of is equicontinuous), and
- possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets such that every bounded subset of is contained in some [3]).
Properties[]
- Let be a DF-space and let be a convex balanced subset of Then is a neighborhood of the origin if and only if for every convex, balanced, bounded subset is a neighborhood of the origin in [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 is either a DF-space or an LM-space. If 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 is a complete nuclear DF-space then is a Montel space.[6]
Sufficient conditions[]
The strong dual space of a Fréchet space 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 truee[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 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 and 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[]
- ^ Jump up to: a b c d e Schaefer & Wolff 1999, pp. 154–155.
- ^ Schaefer & Wolff 1999, pp. 152, 154.
- ^ Schaefer & Wolff 1999, p. 25.
- ^ Schaefer & Wolff 1999, p. 196.
- ^ Schaefer & Wolff 1999, pp. 190–202.
- ^ Jump up to: a b Schaefer & Wolff 1999, pp. 199–202.
- ^ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
- ^ Jump up to: a b Schaefer & Wolff 1999, p. 154.
- ^ Khaleelulla 1982, p. 33.
- ^ Jump up to: a b c d e f g h Schaefer & Wolff 1999, pp. 196–197.
- ^ 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.
- (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 66 (Second ed.). Berlin, New York: . 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.; (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- (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[]
- Topology
- Topological vector spaces
- Functional analysis