DF-space explained

In the mathematical 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.

DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in . 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 V _{1,} V _{2,} \ldots is a sequence of convex 0-neighborhoods in X ^{\prime} _{b} such that V : = \cap _{i} V _{i} absorbs every strongly bounded set, then V is a 0-neighborhood in X ^{\prime} _{b} (where X ^{\prime} _{b} is the continuous dual space of X endowed with the strong dual topology).

Definition

A locally convex topological vector space (TVS)

X is a DF-space, also written (DF)-space, ifX is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of X ^{\prime} is equicontinuous), andX possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets B _{1,} B _{2,} \ldots such that every bounded subset of X is contained in some B _{i} ).

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 \subseteq X, B \cap V is a neighborhood of the origin in B. 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.
• The strong dual space of a DF-space is a Fréchet space.
• Every infinite-dimensional Montel DF-space is a sequential space but 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.
• If X is a complete nuclear DF-space then X is a Montel space.

Sufficient conditions

X ^{\prime} _{b} of a Fréchet space X is a DF-space.^[1]

• The strong dual of a metrizable locally convex space is a DF-space but the convers is in general not true (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.
  • Every Banach space is a DF-space.
  • 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.
• The completion of a DF-space is a DF-space.
• The locally convex sum of a sequence of DF-spaces is a DF-space.
• An inductive limit of a sequence of DF-spaces is a DF-space.
• 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.

However,

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

Examples

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

Bibliography

• Grothendieck . Alexander . Alexander Grothendieck . fr . Sur les espaces (F) et (DF) . Summa Brasil. Math. . 3 . 1954 . 75542 . 57–123.
  • • • Book: Pietsch, Albrecht. Nuclear locally convex spaces. Springer-Verlag. Berlin, New York. 1972. 0-387-05644-0. 539541 .

External links

• DF-space at ncatlab

Notes and References

1. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)