View source for DF-space

In the [[mathematics|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 space]]s. 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 <math>X</math> is a [[Metrizable topological vector space|metrizable]] locally convex space and <math>V_1, V_2, \ldots</math> is a sequence of convex 0-neighborhoods in <math>X^{\prime}_b</math> such that <math>V := \cap_{i} V_i</math> absorbs every strongly bounded set, then <math>V</math> is a 0-neighborhood in <math>X^{\prime}_b</math> (where <math>X^{\prime}_b</math> is the continuous dual space of <math>X</math> endowed with the strong dual topology).{{sfn|Schaefer|Wolff|1999|pp=152,154}}

== Definition ==

A [[locally convex topological vector space]] (TVS) <math>X</math> is a '''DF-space''', also written '''(''DF'')-space''', if{{sfn|Schaefer|Wolff|1999|pp=154-155}}

# <math>X</math> is a [[countably quasi-barrelled space]] (i.e. every strongly bounded countable union of equicontinuous subsets of <math>X^{\prime}</math> is equicontinuous), and
# <math>X</math> possesses a fundamental sequence of bounded (i.e. there exists a countable sequence of bounded subsets <math>B_1, B_2, \ldots</math> such that every bounded subset of <math>X</math> is contained in some <math>B_i</math>{{sfn|Schaefer|Wolff|1999|p=25}}).

== Properties ==

<ul>
<li>Let <math>X</math> be a DF-space and let <math>V</math> be a convex balanced subset of <math>X.</math> Then <math>V</math> is a neighborhood of the origin if and only if for every convex, balanced, bounded subset <math>B \subseteq X,</math> <math>B \cap V</math> is a neighborhood of the origin in <math>B.</math>{{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}}</li>
<li>The [[strong dual space]] of a DF-space is a [[Fréchet space]].{{sfn|Schaefer|Wolff|1999|p=196}}</li>
<li>Every infinite-dimensional [[Montel space|Montel]] DF-space is a [[sequential space]] but {{em|not}} a [[Fréchet–Urysohn space]].</li>
<li>Suppose <math>X</math> is either a DF-space or an [[LM-space]]. If <math>X</math> is a [[sequential space]] then it is either [[Metrizable topological vector space|metrizable]] or else a [[Montel space]] DF-space.</li>
<li>Every [[quasi-complete]] DF-space is complete.{{sfn|Schaefer|Wolff|1999|pp=190-202}}</li>
<li>If <math>X</math> is a [[Complete topological vector space|complete]] [[Nuclear space|nuclear]] DF-space then <math>X</math> is a [[Montel space]].{{sfn|Schaefer|Wolff|1999|pp=199-202}}</li>
</ul>

== Sufficient conditions ==

The [[strong dual space]] <math>X_b^{\prime}</math> of a Fréchet space <math>X</math> is a DF-space.<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref>

<ul>
<li>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.</li>
<li>Every Hausdorff quotient of a DF-space is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}</li>
<li>The [[Complete topological vector space|completion]] of a DF-space is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}</li>
<li>The locally convex sum of a sequence of DF-spaces is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}</li>
<li>An inductive limit of a sequence of DF-spaces is a DF-space.{{sfn|Schaefer|Wolff|1999|pp=196-197}}</li>
<li>Suppose that <math>X</math> and <math>Y</math> 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}}<li>
</ul>

However,

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

== 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}}

== See also ==

* {{annotated link|Barreled space}}
* {{annotated link|Countably quasi-barrelled space}}
* {{annotated link|F-space}}
* {{annotated link|LB-space}}
* {{annotated link|LF-space}}
* {{annotated link|Nuclear space}}
* {{annotated link|Projective tensor product}}

== Citations ==
{{reflist|group=note}}
{{reflist}}

== 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}} <!-- {{sfn|Grothendieck|1954|p=}} -->
* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} <!-- {{sfn|Grothendieck|1955|p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} --> 
* {{Pietsch Nuclear Locally Convex Spaces|edition=2}} <!-- {{sfn|Pietsch|1979|p=}} -->
* {{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 }} <!-- {{sfn|Pietsch|1972|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!-- {{sfn|Wong|1979|p=}} -->

== External links ==

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

{{Functional analysis}}
{{TopologicalVectorSpaces}}

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[[Category:Topology]]
[[Category:Topological vector spaces]]
[[Category:Functional analysis]]