View source for Barrelled space

{{Short description|Type of topological vector space}}
In [[functional analysis]] and related areas of [[mathematics]], a '''barrelled space''' (also written '''barreled space''') is a [[topological vector space]] (TVS) for which every [[barrelled set]] in the space is a [[Neighbourhood (topology)|neighbourhood]] for the [[zero vector]]. 
A '''barrelled set''' or a '''barrel''' in a topological vector space is a [[Set (mathematics)|set]] that is [[Convex set|convex]], [[Balanced set|balanced]], [[Absorbing set|absorbing]], and [[Closed set|closed]]. 
Barrelled spaces are studied because a form of the [[Banach–Steinhaus theorem]] still holds for them. 
Barrelled spaces were introduced by {{harvs|last=Bourbaki|authorlink=Nicolas Bourbaki|year=1950|txt}}. 

==Barrels==

A [[Convex set|convex]] and [[Balanced set|balanced]] [[Set (mathematics)|subset]] of a real or complex vector space is called a {{em|disk}} and it is said to be {{em|disked}}, {{em|[[Absolutely convex set|absolutely convex]]}}, or {{em|convex balanced}}.

A '''{{em|{{visible anchor|barrel}}}}''' or a '''{{em|[[Barrelled set|{{visible anchor|barrelled set}}]]}}''' in a [[topological vector space]] (TVS) is a subset that is a [[Closed set|closed]] [[Absorbing set|absorbing]] disk; that is, a barrel is a convex, balanced, closed, and absorbing subset.

Every barrel must contain the origin. If <math>\dim X \geq 2</math> and if <math>S</math> is any subset of <math>X,</math> then <math>S</math> is a convex, balanced, and absorbing set of <math>X</math> if and only if this is all true of <math>S \cap Y</math> in <math>Y</math> for every <math>2</math>-dimensional vector subspace <math>Y;</math> thus if <math>\dim X > 2</math> then the requirement that a barrel be a [[Closed set|closed subset]] of <math>X</math> is the only defining property that does not depend {{em|solely}} on <math>2</math> (or lower)-dimensional vector subspaces of <math>X.</math> 

If <math>X</math> is any TVS then every closed convex and balanced [[Neighbourhood (mathematics)|neighborhood]] of the origin is necessarily a barrel in <math>X</math> (because every neighborhood of the origin is necessarily an absorbing subset). In fact, every [[locally convex topological vector space]] has a [[neighborhood basis]] at its origin consisting entirely of barrels. However, in general, there {{em|might}} exist barrels that are not neighborhoods of the origin; "barrelled spaces" are exactly those TVSs in which every barrel is necessarily a neighborhood of the origin. Every finite dimensional topological vector space is a barrelled space so examples of barrels that are not neighborhoods of the origin can only be found in infinite dimensional spaces. 

===Examples of barrels and non-barrels===

The closure of any convex, balanced, and absorbing subset is a barrel. This is because the closure of any convex (respectively, any balanced, any absorbing) subset has this same property. 

''A family of examples'': Suppose that <math>X</math> is equal to <math>\Complex</math> (if considered as a complex vector space) or equal to <math>\R^2</math> (if considered as a real vector space). Regardless of whether <math>X</math> is a real or complex vector space, every barrel in <math>X</math> is necessarily a neighborhood of the origin (so <math>X</math> is an example of a barrelled space). Let <math>R : [0, 2\pi) \to (0, \infty]</math> be any function and for every angle <math>\theta \in [0, 2 \pi),</math> let <math>S_{\theta}</math> denote the closed line segment from the origin to the point <math>R(\theta) e^{i \theta} \in \Complex.</math> Let <math display="inline">S := \bigcup_{\theta \in [0, 2 \pi)} S_{\theta}.</math> Then <math>S</math> is always an absorbing subset of <math>\R^2</math> (a real vector space) but it is an absorbing subset of <math>\Complex</math> (a complex vector space) if and only if it is a [[Neighbourhood (mathematics)|neighborhood]] of the origin. Moreover, <math>S</math> is a balanced subset of <math>\R^2</math> if and only if <math>R(\theta) = R(\pi + \theta)</math> for every <math>0 \leq \theta < \pi</math> (if this is the case then <math>R</math> and <math>S</math> are completely determined by <math>R</math>'s values on <math>[0, \pi)</math>) but <math>S</math> is a balanced subset of <math>\Complex</math> if and only it is an open or closed ball centered at the origin (of radius <math>0 < r \leq \infty</math>). In particular, barrels in <math>\Complex</math> are exactly those closed balls centered at the origin with radius in <math>(0, \infty].</math> If <math>R(\theta) := 2 \pi - \theta</math> then <math>S</math> is a closed subset that is absorbing in <math>\R^2</math> but not absorbing in <math>\Complex,</math> and that is neither convex, balanced, nor a neighborhood of the origin in <math>X.</math> By an appropriate choice of the function <math>R,</math> it is also possible to have <math>S</math> be a balanced and absorbing subset of <math>\R^2</math> that is neither closed nor convex. To have <math>S</math> be a balanced, absorbing, and closed subset of <math>\R^2</math> that is {{em|neither}} convex nor a neighborhood of the origin, define <math>R</math> on <math>[0, \pi)</math> as follows: for <math>0 \leq \theta < \pi,</math> let <math>R(\theta) := \pi - \theta</math> (alternatively, it can be any positive function on <math>[0, \pi)</math> that is continuously differentiable, which guarantees that <math display="inline">\lim_{\theta \searrow 0} R(\theta) = R(0) > 0</math> and that <math>S</math> is closed, and that also satisfies <math display="inline">\lim_{\theta \nearrow \pi} R(\theta) = 0,</math> which prevents <math>S</math> from being a neighborhood of the origin) and then extend <math>R</math> to <math>[\pi, 2 \pi)</math> by defining <math>R(\theta) := R(\theta - \pi),</math> which guarantees that <math>S</math> is balanced in <math>\R^2.</math>

===Properties of barrels===

<ul>
<li>In any [[topological vector space]] (TVS) <math>X,</math> every barrel in <math>X</math> [[Absorbing set|absorbs]] every compact convex subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
<li>In any [[Locally convex topological vector space|locally convex]] Hausdorff TVS <math>X,</math> every barrel in <math>X</math> absorbs every convex bounded complete subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
<li>If <math>X</math> is locally convex then a subset <math>H</math> of <math>X^{\prime}</math> is <math>\sigma\left(X^{\prime}, X\right)</math>-bounded if and only if there exists a [[#barrel|barrel]] <math>B</math> in <math>X</math> such that <math>H \subseteq B^{\circ}.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
<li>Let <math>(X, Y, b)</math> be a [[Dual system|pairing]] and let <math>\nu</math> be a locally convex topology on <math>X</math> consistent with duality. Then a subset <math>B</math> of <math>X</math> is a barrel in <math>(X, \nu)</math> if and only if <math>B</math> is the [[Polar set|polar]] of some <math>\sigma(Y, X, b)</math>-bounded subset of <math>Y.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
<li>Suppose <math>M</math> is a vector subspace of finite codimension in a locally convex space <math>X</math> and <math>B \subseteq M.</math> If <math>B</math> is a barrel (resp. [[Bornivorous set|bornivorous]] barrel, bornivorous disk) in <math>M</math> then there exists a barrel (resp. bornivorous barrel, bornivorous disk) <math>C</math> in <math>X</math> such that <math>B = C \cap M.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
</ul>

==Characterizations of barreled spaces==

Denote by <math>L(X; Y)</math> the space of continuous linear maps from <math>X</math> into <math>Y.</math>

If <math>(X, \tau)</math> is a [[Hausdorff space|Hausdorff]] [[topological vector space]] (TVS) with [[continuous dual space]] <math>X^{\prime}</math> then the following are equivalent:

<ol>
<li><math>X</math> is barrelled.</li>
<li>'''{{em|Definition}}''': Every barrel in <math>X</math> is a neighborhood of the origin.
* This definition is similar to a characterization of Baire TVSs proved by Saxon [1974], who proved that a TVS <math>Y</math> with a topology that is not the [[indiscrete topology]] is a [[Baire space]] if and only if every absorbing balanced subset is a neighborhood of {{em|some}} point of <math>Y</math> (not necessarily the origin).{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>For any Hausdorff TVS <math>Y</math> every pointwise bounded subset of <math>L(X; Y)</math> is equicontinuous.{{sfn|Adasch|Ernst|Keim|1978|p=39}}</li>
<li>For any [[F-space]] <math>Y</math> every pointwise bounded subset of <math>L(X; Y)</math> is equicontinuous.{{sfn|Adasch|Ernst|Keim|1978|p=39}}
* An F-space is a complete [[Metrizable topological vector space|metrizable TVS]].</li>
<li>Every [[closed linear operator]] from <math>X</math> into a complete metrizable TVS is continuous.{{sfn|Adasch|Ernst|Keim|1978|p=43}}
* A linear map <math>F : X \to Y</math> is called '''closed''' if its [[Closed graph|graph is a closed]] subset of <math>X \times Y.</math></li>
<li>Every Hausdorff TVS topology <math>\nu</math> on <math>X</math> that has a neighborhood basis of the origin consisting of <math>\tau</math>-closed set is course than <math>\tau.</math>{{sfn|Adasch|Ernst|Keim|1978|p=32}}</li>
</ol>

If <math>(X, \tau)</math> is [[Locally convex topological vector space|locally convex]] space then this list may be extended by appending:
<ol start=7>
<li>There exists a TVS <math>Y</math> not carrying the [[indiscrete topology]] (so in particular, <math>Y \neq \{0\}</math>) such that every pointwise bounded subset of <math>L(X; Y)</math> is equicontinuous.{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>For any locally convex TVS <math>Y,</math> every pointwise bounded subset of <math>L(X; Y)</math> is equicontinuous.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* It follows from the above two characterizations that in the class of locally convex TVS, barrelled spaces are exactly those for which the uniform boundedness principle holds.</li>
<li>Every <math>\sigma\left(X^{\prime}, X\right)</math>-bounded subset of the continuous dual space <math>X</math> is equicontinuous (this provides a partial converse to the [[Banach-Steinhaus theorem]]).{{sfn|Narici|Beckenstein|2011|pp=371-423}}<ref name="Schaefer (1999) p. 127, 141, Trèves (2006) p. 350">{{harvnb|Schaefer|Wolff|1999|pp=127, 141}}{{harvnb|Trèves|2006|p=350}}.</ref></li>
<li><math>X</math> carries the [[strong dual topology]] <math>\beta\left(X, X^{\prime}\right).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>Every [[lower semicontinuous]] [[seminorm]] on <math>X</math> is continuous.{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>Every linear map <math>F : X \to Y</math> into a locally convex space <math>Y</math> is [[almost continuous]].{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* A linear map <math>F : X \to Y</math> is called '''{{em|{{visible anchor|almost continuous}}}}''' if for every neighborhood <math>V</math> of the origin in <math>Y,</math> the closure of <math>F^{-1}(V)</math> is a neighborhood of the origin in <math>X.</math></li>
<li>Every surjective linear map <math>F : Y \to X</math> from a locally convex space <math>Y</math> is [[Almost open map|almost open]].{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* This means that for every neighborhood <math>V</math> of 0 in <math>Y,</math> the closure of <math>F(V)</math> is a neighborhood of 0 in <math>X.</math></li>
<li>If <math>\omega</math> is a locally convex topology on <math>X</math> such that <math>(X, \omega)</math> has a neighborhood basis at the origin consisting of <math>\tau</math>-closed sets, then <math>\omega</math> is weaker than <math>\tau.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
</ol>

If <math>X</math> is a Hausdorff locally convex space then this list may be extended by appending:
<ol start=15>
<li>'''[[Closed graph theorem]]''': Every [[closed linear operator]] <math>F : X \to Y</math> into a [[Banach space]] <math>Y</math> is [[Continuous linear operator|continuous]].{{sfn|Narici|Beckenstein|2011|p=477}}
* The linear operator is called {{em|closed}} if its [[Closed graph|graph is a closed subset]] of <math>X \times Y.</math></li>
<li>For every subset <math>A</math> of the continuous dual space of <math>X,</math> the following properties are equivalent: <math>A</math> is<ref name="Schaefer (1999) p. 127, 141, Trèves (2006) p. 350"/>
<ol style="list-style-type: lower-roman;">
<li>equicontinuous;</li>
<li>relatively weakly compact;</li>
<li>strongly bounded;</li>
<li>weakly bounded.</li>
</ol></li>
<li>The 0-neighborhood bases in <math>X</math> and the fundamental families of bounded sets in <math>X_{\beta}^{\prime}</math> correspond to each other by [[Polar set|polarity]].<ref name="Schaefer (1999) p. 127, 141, Trèves (2006) p. 350"/></li>
</ol>

If <math>X</math> is [[metrizable topological vector space]] then this list may be extended by appending:
<ol start=18>
<li>For any complete metrizable TVS <math>Y</math> every pointwise bounded {{em|sequence}} in <math>L(X; Y)</math> is equicontinuous.{{sfn|Adasch|Ernst|Keim|1978|p=39}}</li>
</ol>

If <math>X</math> is a locally convex [[metrizable topological vector space]] then this list may be extended by appending:
<ol start=19>
<li>({{visible anchor|Property S}}): The [[weak* topology]] on <math>X^{\prime}</math> is [[sequentially complete]].{{sfn|Narici|Beckenstein|2011|p=399}}</li>
<li>({{visible anchor|Property C}}): Every weak* bounded subset of <math>X^{\prime}</math> is <math>\sigma\left(X^{\prime}, X\right)</math>-relatively [[countably compact]].{{sfn|Narici|Beckenstein|2011|p=399}}</li>
<li>({{visible anchor|𝜎-barrelled}}): Every countable weak* bounded subset of <math>X^{\prime}</math> is equicontinuous.{{sfn|Narici|Beckenstein|2011|p=399}}</li>
<li>({{visible anchor|Baire-like}}): <math>X</math> is not the union of an increase sequence of [[Nowhere dense set|nowhere dense]] [[Absolutely convex set|disks]].{{sfn|Narici|Beckenstein|2011|p=399}}</li>
</ol>

==Examples and sufficient conditions==

Each of the following topological vector spaces is barreled:
<ol>
<li>TVSs that are [[Baire space]].
* Consequently, every topological vector space that is of the [[second category]] in itself is barrelled.</li>
<li>[[F-space]]s, [[Fréchet space]]s, [[Banach space]]s, and [[Hilbert space]]s.
* However, there exist [[normed vector space]]s that are {{em|not}} barrelled. For example, if the [[Lp space|<math>L^p</math>-space]] <math>L^2([0, 1])</math> is topologized as a subspace of <math>L^1([0, 1]),</math> then it is not barrelled.</li>
<li>[[Complete topological vector space|Complete]] [[Metrizable topological vector space|pseudometrizable]] TVSs.{{sfn|Narici|Beckenstein|2011|p=383}}
* Consequently, every finite-dimensional TVS is barrelled.</li>
<li>[[Montel space]]s.</li>
<li>[[Strong dual space]]s of Montel spaces (since they are necessarily Montel spaces).</li>
<li>A locally convex [[quasi-barrelled space]] that is also a [[σ-barrelled space]].{{sfn|Khaleelulla|1982|pp=28-63}}</li>
<li>A sequentially complete [[quasibarrelled space]].</li>
<li>A [[Quasi-complete space|quasi-complete]] Hausdorff locally convex [[infrabarrelled space]].{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* A TVS is called '''quasi-complete''' if every closed and bounded subset is complete.</li>
<li>A TVS with a dense barrelled vector subspace.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* Thus the completion of a barreled space is barrelled.</li>
<li>A Hausdorff locally convex TVS with a dense [[Infrabarrelled space|infrabarrelled]] vector subspace.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>A vector subspace of a barrelled space that has countable codimensional.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
* In particular, a finite codimensional vector subspace of a barrelled space is barreled.</li>
<li>A locally convex ultrabarelled TVS.{{sfn|Narici|Beckenstein|2011|pp=418-419}}</li>
<li>A Hausdorff locally convex TVS <math>X</math> such that every weakly bounded subset of its continuous dual space is equicontinuous.{{sfn|Trèves|2006|p=350}}</li>
<li>A locally convex TVS <math>X</math> such that for every Banach space <math>B,</math> a closed linear map of <math>X</math> into <math>B</math> is necessarily continuous.{{sfn|Schaefer|Wolff|1999|p=166}}</li>
<li>A product of a family of barreled spaces.{{sfn|Schaefer|Wolff|1999|p=138}}</li>
<li>A locally convex direct sum and the inductive limit of a family of barrelled spaces.{{sfn|Schaefer|Wolff|1999|p=61}}</li>
<li>A quotient of a barrelled space.{{sfn|Trèves|2006|p=346}}{{sfn|Schaefer|Wolff|1999|p=61}}</li>
<li>A Hausdorff [[sequentially complete]] [[quasibarrelled]] boundedly summing TVS.{{sfn|Adasch|Ernst|Keim|1978|p=77}}</li>
<li>A locally convex Hausdorff [[reflexive space]] is barrelled.</li>
</ol>

===Counter examples===

<ul>
<li>A barrelled space need not be [[Montel space|Montel]], [[Complete topological vector space|complete]], [[Metrizable topological vector space|metrizable]], unordered Baire-like, nor the inductive limit of Banach spaces.</li>
<li>Not all normed spaces are barrelled. However, they are all infrabarrelled.{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>A closed subspace of a barreled space is not necessarily [[Countably quasi-barrelled space|countably quasi-barreled]] (and thus not necessarily barrelled).{{sfn|Schaefer|Wolff|1999|pp=103-110}}</li>
<li>There exists a dense vector subspace of the [[Fréchet space|Fréchet]] barrelled space <math>\R^{\N}</math> that is not barrelled.{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>There exist complete locally convex TVSs that are not barrelled.{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>The [[Locally convex topological vector space#Finest and coarsest locally convex topology|finest locally convex topology]] on an infinite-dimensional vector space is a Hausdorff barrelled space that is a [[Meager set|meagre]] subset of itself (and thus not a [[Baire space]]).{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
</ul>

==Properties of barreled spaces==

===Banach–Steinhaus generalization===

The importance of barrelled spaces is due mainly to the following results.

{{Math theorem|name=Theorem{{sfn|Trèves|2006|p=347}}|math_statement=
Let <math>X</math> be a barrelled TVS and <math>Y</math> be a locally convex TVS. 
Let <math>H</math> be a subset of the space <math>L(X ;Y)</math> of continuous linear maps from <math>X</math> into <math>Y</math>. 
The following are equivalent:
<ol>
<li><math>H</math> is bounded for the topology of pointwise convergence;</li>
<li><math>H</math> is bounded for the topology of bounded convergence;</li>
<li><math>H</math> is [[equicontinuous]].</li>
</ol>
}}

The [[Banach-Steinhaus theorem]] is a corollary of the above result.{{sfn|Trèves|2006|p=348}} When the vector space <math>Y</math> consists of the complex numbers then the following generalization also holds.

{{Math theorem|name=Theorem{{sfn|Trèves|2006|p=349}}|math_statement=
If <math>X</math> is a barrelled TVS over the complex numbers and <math>H</math> is a subset of the continuous dual space of <math>X</math>, then the following are equivalent:
<ol>
<li><math>H</math> is weakly bounded;</li>
<li><math>H</math> is strongly bounded;</li>
<li><math>H</math> is equicontinuous;</li>
<li><math>H</math> is relatively compact in the weak dual topology.</li>
</ol>
}}

Recall that a linear map <math>F : X \to Y</math> is called '''closed''' if its graph is a closed subset of <math>X \times Y.</math>

{{Math theorem|name=Closed Graph Theorem{{sfn|Adasch|Ernst|Keim|1978|p=41}}|math_statement=
Every closed linear operator from a Hausdorff barrelled TVS into a complete metrizable TVS is continuous. 
}}

===Other properties===

<ul>
<li>Every Hausdorff barrelled space is [[quasi-barrelled]].{{sfn|Adasch|Ernst|Keim|1978|pp=70-73}}</li>
<li>A linear map from a barrelled space into a locally convex space is [[#almost continuous|almost continuous]].</li>
<li>A linear map from a locally convex space {{em|on}}to a barrelled space is [[Almost open map|almost open]].</li>
<li>A [[separately continuous]] bilinear map from a product of barrelled spaces into a locally convex space is [[hypocontinuous]].{{sfn|Trèves|2006|p=424}}</li>
<li>A linear map with a closed graph from a barreled TVS into a <math>B_r</math>-complete TVS is necessarily continuous.{{sfn|Schaefer|Wolff|1999|p=166}}</li>
</ul>

==See also==

* {{annotated link|Barrelled set}}
* {{annotated link|Countably barrelled space}}
* {{annotated link|Distinguished space}}
* {{annotated link|Quasibarrelled space}}
* {{annotated link|Ultrabarrelled space}}
* {{annotated link|Uniform boundedness principle#Generalisations}}
* {{annotated link|Ursescu theorem}}
* {{annotated link|Webbed space}}

==References==

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

==Bibliography==

{{Refbegin}}
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* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
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* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1969|p=}} -->
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* {{Robertson Topological Vector Spaces}} <!-- {{sfn|Robertson|Robertson|1980|p=}} -->
* {{cite book|last1=Robertson|first1=Alex P.|last2=Robertson|first2= Wendy J.|title= Topological vector spaces|series=Cambridge Tracts in Mathematics|volume=53|year=1964|publisher= [[Cambridge University Press]]|pages=65–75}}
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* {{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=}} -->
{{Refend}}
* {{Voigt A Course on Topological Vector Spaces|edition=1}} <!-- {{sfn|Voigt|2020|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional analysis}}
{{Boundedness and bornology}}
{{Topological vector spaces}}

[[Category:Topological vector spaces]]