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{{short description|Barrelled space where closed and bounded subsets are compact}}
In [[Functional analysis|functional analysis]] and related areas of [[Mathematics|mathematics]], a '''Montel space''', named after Paul Montel, is any [[Topological vector space|topological vector space]] (TVS) in which an analog of [[Montel's theorem]] holds.  Specifically, a Montel space is a [[Barrelled space|barrelled]] topological vector space in which every [[Closed set|closed]] and [[Bounded set (topological vector space)|bounded subset]] is [[Compact space|compact]]. 

==Definition==

A [[Topological vector space|topological vector space]] (TVS) has the '''[[Heine–Borel theorem#In the theory of topological vector spaces|{{visible anchor|Heine–Borel property}}]]''' if every [[Closed set|closed]] and [[Bounded set (topological vector space)|bounded subset]] is [[Compact space|compact]]. 
A '''{{visible anchor|Montel space}}''' is a [[Barrelled space|barrelled]] topological vector space with the Heine–Borel property. Equivalently, it is an [[Infrabarrelled space|infrabarrelled]] semi-Montel space where a [[Hausdorff space|Hausdorff]] [[Locally convex topological vector space|locally convex topological vector space]] is called a '''{{visible anchor|semi-Montel space}}''' or '''{{visible anchor|perfect}}''' if every [[Bounded set (topological vector space)|bounded subset]] is relatively compact.<ref group=note>A subset <math>S</math> of a topological space <math>X</math> is called '''relatively compact''' is its closure in <math>X</math> is [[Compact space|compact]].</ref> 
A subset of a TVS is compact if and only if it is [[Complete topological vector space#complete subset|complete]] and [[Totally bounded space|totally bounded]].
A '''{{visible anchor|Fréchet–Montel space}}''' is a [[Fréchet space]] that is also a Montel space. 

==Characterizations==

A [[Separable space|separable]] [[Fréchet space]] is a Montel space if and only if each weak-* convergent sequence in its continuous dual is [[Strong dual space|strongly convergent]].{{sfn|Schaefer|Wolff|1999|pp=194-195}}

A [[Fréchet space]] <math>X</math> is a Montel space if and only if every bounded continuous function <math>X \to c_0</math> sends closed bounded absolutely convex subsets of <math>X</math> to relatively compact subsets of <math>c_0.</math> 
Moreover, if <math>C^b(X)</math> denotes the vector space of all bounded continuous functions on a [[Fréchet space]] <math>X,</math> then <math>X</math> is Montel if and only if every sequence in <math>C^b(X)</math> that converges to zero in the [[Compact-open topology|compact-open topology]] also converges uniformly to zero on all closed bounded absolutely convex subsets of <math>X.</math>
{{sfn|Lindström|1990|pp=191–196}}

==Sufficient conditions==

'''Semi-Montel spaces'''

A closed vector subspace of a semi-Montel space is again a semi-Montel space.  The locally convex [[Direct sum|direct sum]] of any family of semi-Montel spaces is again a semi-Montel space.  The [[Inverse limit|inverse limit]] of an inverse system consisting of semi-Montel spaces is again a semi-Montel space.  The [[Cartesian product]] of any family of semi-Montel spaces (resp. Montel spaces) is again a semi-Montel space (resp. a Montel space).

'''Montel spaces'''

The strong dual of a Montel space is Montel.  
A [[Barrelled space|barrelled]] [[Quasi-complete space|quasi-complete]] [[Nuclear space|nuclear space]] is a Montel space.{{sfn|Schaefer|Wolff|1999|pp=194-195}}  
Every product and locally convex direct sum of a family of Montel spaces is a Montel space.{{sfn|Schaefer|Wolff|1999|pp=194-195}}  
The strict inductive limit of a sequence of Montel spaces is a Montel space.{{sfn|Schaefer|Wolff|1999|pp=194-195}}  In contrast, closed subspaces and separated quotients of Montel spaces are in general not even [[Reflexive space|reflexive]].{{sfn|Schaefer|Wolff|1999|pp=194-195}}  
Every [[Fréchet space|Fréchet]] [[Schwartz topological vector space|Schwartz]] space is a Montel space.{{sfn|Khaleelulla|1982|pp=32-63}}  

==Properties==

Montel spaces are [[Paracompact space|paracompact]] and [[Normal space|normal]].<ref name="Encyc. Math TVS">{{cite web|title=Topological vector space|author=<!--Not stated-->|date=|website=Encyclopedia of Mathematics|access-date=September 6, 2020|url=https://encyclopediaofmath.org/wiki/Topological_vector_space|quote=}}</ref>  
Semi-Montel spaces are [[Quasi-complete space|quasi-complete]] and [[Semi-reflexive space|semi-reflexive]] while Montel spaces are [[Reflexive space|reflexive]].  

No infinite-dimensional [[Banach space]] is a Montel space.  This is because a Banach space cannot satisfy the [[Heine–Borel theorem#In the theory of topological vector spaces|Heine–Borel property]]: the closed unit ball is closed and bounded, but not compact.  
[[Fréchet space|Fréchet]] Montel spaces are separable and have a [[Bornological space|bornological]] strong dual.  
A metrizable Montel space is [[Separable space|separable]].{{sfn|Schaefer|Wolff|1999|pp=194-195}}

Fréchet–Montel spaces are [[Distinguished space|distinguished space]]s.

==Examples==

In classical [[Complex analysis|complex analysis]], Montel's theorem asserts that the space of [[Holomorphic function|holomorphic function]]s on an [[Open set|open]] [[Connected space|connected]] subset of the [[Complex number|complex number]]s has this property.

Many Montel spaces of contemporary interest arise as spaces of test functions for a space of [[Distribution (mathematics)|distributions]]. 
The space <math>C^{\infty}(\Omega)</math> of smooth functions on an open set <math>\Omega</math> in <math>\R^n</math> is a Montel space equipped with the topology induced by the family of [[Seminorm|seminorm]]s
<math display=block>\|f\|_{K,n} = \sup_{|\alpha| \leq n}\sup_{x \in K}\left|\partial^\alpha f(x)\right|</math>
for <math>n = 1, 2, \ldots</math> and <math>K</math> ranges over compact subsets of <math>\Omega,</math> and <math>\alpha</math> is a multi-index.  Similarly, the space of compactly supported functions in an open set with the [[Final topology|final topology]] of the family of inclusions <math>\scriptstyle{C^\infty_0(K)\subset C^\infty_0(\Omega)}</math> as <math>K</math> ranges over all compact subsets of <math>\Omega.</math> The [[Schwartz space]] is also a Montel space.

===Counter-examples===

Every infinite-dimensional normed space is a [[Barrelled space|barrelled space]] that is {{em|not}} a Montel space.{{sfn|Khaleelulla|1982|pp=28-63}} 
In particular, every infinite-dimensional [[Banach space]] is not a Montel space.{{sfn|Khaleelulla|1982|pp=28-63}}  
There exist Montel spaces that are not [[Separable space|separable]] and there exist Montel spaces that are not [[Complete topological vector space|complete]].{{sfn|Khaleelulla|1982|pp=28-63}} 
There exist Montel spaces having closed vector subspaces that are {{em|not}} Montel spaces.{{sfn|Khaleelulla|1982|pp=103-110}}

==See also==

* {{annotated link|Barrelled space}}
* {{annotated link|Bornological space}}
* {{annotated link|Heine–Borel theorem}}
* {{annotated link|LB-space}}
* {{annotated link|LF-space}}
* {{annotated link|Nuclear space}}

==Notes==

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

==References==

{{reflist}}

* {{Edwards Functional Analysis Theory and Applications}} <!--{{sfn|Edwards|1995|p=}}-->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!--{{sfn|Hogbe-Nlend|1977|p=}}-->
* {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}} <!--{{sfn|Hogbe-Nlend|Moscatelli|1981|p=}}-->
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!--{{sfn|Khaleelulla|1982|p=}}-->
* {{Köthe Topological Vector Spaces I}} <!--{{sfn|Köthe|1969|p=}}-->
* {{Köthe Topological Vector Spaces II}} <!--{{sfn|Köthe|1979|p=}}-->
* {{cite journal|last=Lindström|first=Mikael|title=A note on Fréchet-Montel spaces|journal=Proceedings of the American Mathematical Society|publisher=American Mathematical Society (AMS)|volume=108|issue=1|date=1990-01-01|year=1990|issn=0002-9939|doi=10.1090/s0002-9939-1990-0994780-8|pages=191–196|url=https://www.ams.org/journals/proc/1990-108-01/S0002-9939-1990-0994780-8/S0002-9939-1990-0994780-8.pdf}} <!--{{sfn|Lindström|1990|pp=191–196}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Robertson Topological Vector Spaces}} <!--{{sfn|Robertson|Robertson|1980|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
* {{Swartz An Introduction to Functional Analysis}} <!--{{sfn|Swartz|1992|p=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!--{{sfn|Wilansky|2013|p=}}-->
* {{springer|title=Montel space|id=p/m064880}}

{{Functional analysis}}


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

{{Sourceattribution|Montel space}}