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.

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 $X^_b$ such that $V := \cap_ V_i$ absorbs every strongly bounded set, then $V$ is a 0-neighborhood in $X^_b$ (where $X^_b$ 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, if

# $X$ is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of $X^$ is equicontinuous), and
# $X$ 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

The strong dual space $X_b^$ of a Fréchet space $X$ 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.


• 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.

See also

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Citations

Bibliography

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External links

DF-space at ncatlab

{{TopologicalVectorSpaces

Topology
Topological vector spaces
Functional analysis