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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. 
== Definition ==

For a locally convex space ''X'' with continuous dual <math>X^{\prime}</math>, ''X'' is called a '''Schwartz space''' if it satisfies any of the following equivalent conditions:{{sfn | Khaleelulla | {{{year| 1982 }}} | p=32}}
* For every [[Closed set|closed]] [[Convex set|convex]] [[Balanced set|balanced]] neighborhood ''U'' of the origin in ''X'', there exists a neighborhood ''V'' of ''0'' in ''X'' such that for all scalars ''r'' > 0, ''V'' can be covered by finitely many translates of ''rU''. 
* Every bounded subset of ''X'' is totally bounded and for every [[Closed set|closed]] [[Convex set|convex]] [[Balanced set|balanced]] neighborhood ''U'' of the origin in ''X'', there exists a neighborhood ''V'' of ''0'' in ''X'' such that for all scalars ''r'' > 0, there exists a bounded subset ''B'' of ''X'' such that ''V'' &sube; ''B'' + ''rU''. 

== Properties ==

Every [[Fréchet space|Fréchet]] Schwartz space is a [[Montel space]].{{sfn | Khaleelulla | {{{year| 1982 }}} | pp=32-63}}

== Examples ==

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

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

== See also ==

* [[Montel space]]

== References ==
{{Reflist}}

*{{cite journal
 | 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}}
* {{cite book |last1=Robertson |first1=Alex P. |first2= Wendy J.|last2=Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= {{wipe|Cambridge University Press}} | pages=65–75}}
* {{cite book | last=Husain | first=Taqdir | title=Barrelledness in topological and ordered vector spaces | publisher=Springer-Verlag | publication-place=Berlin New York | year=1978 | isbn=3-540-09096-7 | oclc=4493665 | ref=harv}} <!-- {{sfn | Husain | 1978 | p=}} -->
* {{cite book |last=Jarhow |first=Hans |title= Locally convex spaces |year=1981 |publisher= Teubner|ISBN=978-3-322-90561-1| ref = harv}}
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | {{{year| 1982 }}} | p=}} --> 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->

{{Functional Analysis}}
{{BoundednessAndBornology}}
{{TopologicalVectorSpaces}}

[[Category:Topological vector spaces]]

{{Sourceattribution|Schwartz TVS|1}}