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 for the zero vector. A '''barrelled set''' or a '''barrel''' in a topological vector space is a set that is convex, balanced, absorbing, and 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 and balanced subset of a real or complex vector space is called a {{em|disk}} and it is said to be {{em|disked}}, {{em|absolutely convex}}, or {{em|convex balanced}}.

A '''{{em|{{visible anchor|barrel}}}}''' or a '''{{em|{{visible anchor|barrelled set}}}}''' in a topological vector space (TVS) is a subset that is a closed 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 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 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 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> absorbs every compact convex subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li> <li>In any 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 <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 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 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 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 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 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 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 coarser than <math>\tau.</math>{{sfn|Adasch|Ernst|Keim|1978|p=32}}</li> </ol>

If <math>(X, \tau)</math> is 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.{{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.{{sfn|Narici|Beckenstein|2011|p=477}} * The linear operator is called {{em|closed}} if its 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 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 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-spaces, Fréchet spaces, Banach spaces, and Hilbert spaces. * However, there exist normed vector spaces that are {{em|not}} barrelled. For example, if the <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 pseudometrizable TVSs.{{sfn|Narici|Beckenstein|2011|p=383}} * Consequently, every finite-dimensional TVS is barrelled.</li> <li>Montel spaces.</li> <li>Strong dual spaces 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 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 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>

===Counterexamples===

<ul> <li>A barrelled space need not be Montel, complete, 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-barreled (and thus not necessarily barrelled).{{sfn|Schaefer|Wolff|1999|pp=103-110}}</li> <li>There exists a dense vector subspace of the 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 finest locally convex topology on an infinite-dimensional vector space is a Hausdorff barrelled space that is a 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.</li> <li>A linear map from a locally convex space {{em|on}}to a barrelled space is 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==

==Bibliography==

{{Refbegin}} * {{Adasch Topological Vector Spaces|edition=2}}  * {{Berberian Lectures in Functional Analysis and Operator Theory}}  * {{cite journal|last=Bourbaki|first=Nicolas|author-link=Nicolas Bourbaki|journal=Annales de l'Institut Fourier|language=fr|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}}  * {{Conway A Course in Functional Analysis|edition=2}}  * {{Edwards Functional Analysis Theory and Applications}}  * {{Grothendieck Topological Vector Spaces}}  * {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}  * {{Jarchow Locally Convex Spaces}}  * {{Khaleelulla Counterexamples in Topological Vector Spaces}}  * {{Köthe Topological Vector Spaces I}}  * {{Narici Beckenstein Topological Vector Spaces|edition=2}}  * {{Osborne Locally Convex Spaces}}  * {{Robertson Topological Vector Spaces}}  * {{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}} * {{Schaefer Wolff Topological Vector Spaces|edition=2}}  * {{Schechter Handbook of Analysis and Its Foundations}}  * {{Swartz An Introduction to Functional Analysis}}  * {{Trèves François Topological vector spaces, distributions and kernels}}  {{Refend}} * {{Voigt A Course on Topological Vector Spaces|edition=1}}  * {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}

Category:Topological vector spaces