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.Template:Sfn

DF-spaces were first defined by Alexander Grothendieck and studied in detail by him in Template:Harv. 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 $V_1,V_2,\dots$ is a sequence of convex 0-neighborhoods in $X_b^{\prime }$ such that $V:={\cap }_i{V}_i$ absorbs every strongly bounded set, then $V$ is a 0-neighborhood in $X_b^{\prime }$ (where $X_b^{\prime }$ is the continuous dual space of $X$ endowed with the strong dual topology).Template:Sfn

Definition

A locally convex topological vector space (TVS) $X$ is a DF-space, also written (DF)-space, ifTemplate:Sfn

1. $X$ is a countably quasi-barrelled space (i.e. every strongly bounded countable union of equicontinuous subsets of $X^{\prime }$ is equicontinuous), and
2. $X$ possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets $B_1,B_2,\dots$ such that every bounded subset of $X$ is contained in some $B_i$Template:Sfn).

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 $B.$Template:Sfn 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.Template:Sfn
• The strong dual space of a DF-space is a Fréchet space.Template:Sfn
• Every infinite-dimensional Montel DF-space is a sequential space but Template:Em 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.Template:Sfn
• If $X$ is a complete nuclear DF-space then $X$ is a Montel space.Template:Sfn

Sufficient conditions

The strong dual space $X_b^{\prime }$ of a Fréchet space $X$ is a DF-space.

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

However,

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

Examples

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

See also

• Template:Annotated link
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Bibliography

• Template:Cite journal
• Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Pietsch Nuclear Locally Convex Spaces
• Template:Cite book
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products

External links

• DF-space at ncatlab