DF-space

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 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).

Definition

A locally convex topological vector space (TVS) is a DF-space, also written (DF)-space, if

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 ).

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 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 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. If is a complete nuclear DF-space then is a Montel space.

Sufficient conditions

The strong dual space of a Fréchet space is a DF-space.Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)

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. <li>Suppose that and are DF-spaces. Then the projective tensor product, as well as its completion, of these spaces is a DF-space.<li>

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.

DF-space

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