View source for Schwartz topological vector space

In [[Functional analysis|functional analysis]] and related areas of [[Mathematics|mathematics]], '''Schwartz spaces''' are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets. These spaces were introduced by [[Biography:Alexander Grothendieck|Alexander Grothendieck]]. 
==Definition==

A [[Hausdorff space|Hausdorff]] locally convex space {{mvar|X}} with continuous dual <math>X^{\prime}</math>, {{mvar|X}} is called a '''Schwartz space''' if it satisfies any of the following equivalent conditions:{{sfn | Khaleelulla | 1982 | p=32}} 

#For every [[Closed set|closed]] convex balanced neighborhood {{mvar|U}} of the origin in {{mvar|X}}, there exists a neighborhood {{mvar|V}} of {{math|0}} in {{mvar|X}} such that for all real {{math|''r'' > 0}}, {{mvar|V}} can be covered by finitely many translates of {{math|''rU''}}.
#Every bounded subset of {{mvar|X}} is totally bounded and for every [[Closed set|closed]] [[Convex set|convex]] [[Balanced set|balanced]] neighborhood {{mvar|U}} of the origin in {{mvar|X}}, there exists a neighborhood {{mvar|V}} of {{math|0}} in {{mvar|X}} such that for all real {{math|''r'' > 0}}, there exists a bounded subset {{mvar|B}} of {{mvar|X}} such that {{math|''V'' ⊆ ''B'' + ''rU''}}.

==Properties==

Every [[Quasi-complete space|quasi-complete]] Schwartz space is a [[Montel space|semi-Montel space]].  
Every [[Fréchet space|Fréchet]] Schwartz space is a [[Montel space]].{{sfn | Khaleelulla | 1982 | pp=32–63}}  

The [[Strong dual space|strong dual space]] of a [[Complete topological vector space|complete]] Schwartz space is an [[Ultrabornological space|ultrabornological space]].

==Examples and sufficient conditions==

* Vector subspace of Schwartz spaces are Schwartz spaces.
* The quotient of a Schwartz space by a closed vector subspace is again a Schwartz space.
* The [[Cartesian product]] of any family of Schwartz spaces is again a Schwartz space.
* The weak topology induced on a vector space by a family of linear maps valued in Schwartz spaces is a Schwartz space '''''if''''' the weak topology is Hausdorff.
* The locally convex strict inductive limit of any countable sequence of Schwartz spaces (with each space TVS-embedded in the next space) is again a Schwartz space.

===Counter-examples===

Every infinite-dimensional normed space is ''not'' a Schwartz space.{{sfn | Khaleelulla | 1982 | pp=32-63}}  

There exist [[Fréchet space]]s that are not Schwartz spaces and there exist Schwartz spaces that are not [[Montel space]]s.{{sfn | Khaleelulla | 1982 | pp=32-63}}  

==See also==

* {{annotated link|Auxiliary normed space}}
* {{annotated link|Montel space}}

==References==

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

==Bibliography==

* {{cite journal|last=Bourbaki|first=Nicolas|journal=[[Annales de l'Institut Fourier]]|language=French|mr=0042609|pages=5–16 (1951)|title=Sur certains espaces vectoriels topologiques|url=http://www.numdam.org/item?id=AIF_1950__2__5_0|volume=2|year=1950|doi=10.5802/aif.16|doi-access=free}}
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!-- {{sfn | Bourbaki | 1987 | p=}} -->
* {{Robertson Topological Vector Spaces}} <!-- {{sfn | Robertson | Robertson | 1980 | p=}} -->
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!-- {{sfn | Husain | Khaleelulla | 1978 | p=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn | Jarchow | 1981 | p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | 1982 | p=}} --> 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | Wolff | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->

{{Functional analysis}}

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

{{Sourceattribution|Schwartz topological vector space}}