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.{{sfn|Schaefer|Wolff|1999|pp=154-155}}

DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in {{harv|Grothendieck|1954}}.

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).{{sfn|Schaefer|Wolff|1999|pp=152,154}}

Definition

A locally convex topological vector space (TVS) $X$ is a DF-space, also written (DF)-space, if{{sfn|Schaefer|Wolff|1999|pp=154-155}}

$X$ is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of $X^{\prime}$ 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 $B_i$ {{sfn|Schaefer|Wolff|1999|p=25}}).

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 $B.$ {{sfn|Schaefer|Wolff|1999|pp=154-155}} 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.{{sfn|Schaefer|Wolff|1999|pp=154-155}}

The strong dual space of a DF-space is a Fréchet space.{{sfn|Schaefer|Wolff|1999|p=196}}

Every infinite-dimensional Montel DF-space is a sequential space but {{em|not}} 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.{{sfn|Schaefer|Wolff|1999|pp=190-202}}

If $X$ is a complete nuclear DF-space then $X$ is a Montel space.{{sfn|Schaefer|Wolff|1999|pp=199-202}}

Sufficient conditions

The strong dual space $X_b^{\prime}$ of a Fréchet space $X$ is a DF-space.Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "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{{sfn|Schaefer|Wolff|1999|p=154}} but the convers is in general not true{{sfn|Schaefer|Wolff|1999|p=154}} (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.{{sfn|Khaleelulla|1982|p=33}}
- Every Banach space is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=154-155}}
- 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.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

The completion of a DF-space is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

The locally convex sum of a sequence of DF-spaces is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

An inductive limit of a sequence of DF-spaces is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

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.{{sfn|Schaefer|Wolff|1999|pp=199-202}}

However,

An infinite product of non-trivial DF-spaces (i.e. all factors have non-0 dimension) is {{em|not}} a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

A closed vector subspace of a DF-space is not necessarily a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

There exist complete DF-spaces that are not TVS-isomorphic to the strong dual of a metrizable locally convex TVS.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

Examples

There exist complete DF-spaces that are not TVS-isomorphic with the strong dual of a metrizable locally convex space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}

There exist DF-spaces having closed vector subspaces that are not DF-spaces.{{sfn|Khaleelulla|1982|pp=103-110}}

Citations

Bibliography

{{cite journal |last=Grothendieck |first=Alexander |author-link=Alexander Grothendieck |language=fr |title=Sur les espaces (F) et (DF) |journal=Summa Brasil. Math. |volume=3 |year=1954 |mr=75542 |pages=57–123}}
{{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}}
{{Khaleelulla Counterexamples in Topological Vector Spaces}}
{{Pietsch Nuclear Locally Convex Spaces|edition=2}}
{{cite book|last=Pietsch|first=Albrecht|title=Nuclear locally convex spaces|publisher=Springer-Verlag|location=Berlin, New York|year=1972|isbn=0-387-05644-0|oclc=539541 }}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}}

External links

[https://ncatlab.org/nlab/show/DF+space DF-space at ncatlab]

Category:Topology

Category:Topological vector spaces

Category:Functional analysis