Infunctional analysis and related areas ofmathematics, abarrelled space (also writtenbarreled space) is atopological vector space (TVS) for which everybarrelled set in the space is aneighbourhood for thezero vector. Abarrelled set or abarrel in a topological vector space is aset that isconvex,balanced,absorbing, andclosed. Barrelled spaces are studied because a form of theBanach–Steinhaus theorem still holds for them. Barrelled spaces were introduced byBourbaki (1950).

Aconvex andbalancedsubset of a real or complex vector space is called adisk and it is said to bedisked,absolutely convex, orconvex balanced.

Abarrel or abarrelled set in atopological vector space (TVS) is a subset that is aclosedabsorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset.

Every barrel must contain the origin. If${\displaystyle \dim X\ge 2}$ and if${\displaystyle S}$ is any subset of${\displaystyle X,}$ then${\displaystyle S}$ is a convex, balanced, and absorbing set of${\displaystyle X}$ if and only if this is all true of${\displaystyle S\cap Y}$ in${\displaystyle Y}$ for every${\displaystyle 2}$-dimensional vector subspace${\displaystyle Y;}$ thus if${\displaystyle \dim X>2}$ then the requirement that a barrel be aclosed subset of${\displaystyle X}$ is the only defining property that does not dependsolely on${\displaystyle 2}$ (or lower)-dimensional vector subspaces of${\displaystyle X.}$

If${\displaystyle X}$ is any TVS then every closed convex and balancedneighborhood of the origin is necessarily a barrel in${\displaystyle X}$ (because every neighborhood of the origin is necessarily an absorbing subset). In fact, everylocally convex topological vector space has aneighborhood basis at its origin consisting entirely of barrels. However, in general, theremight 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.

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

• In anytopological vector space (TVS)${\displaystyle X,}$ every barrel in${\displaystyle X}$absorbs every compact convex subset of${\displaystyle X.}$^[1]
• In anylocally convex Hausdorff TVS${\displaystyle X,}$ every barrel in${\displaystyle X}$ absorbs every convex bounded complete subset of${\displaystyle X.}$^[1]
• If${\displaystyle X}$ is locally convex then a subset${\displaystyle H}$ of${\displaystyle X^{\mathrm{\prime }}}$ is${\displaystyle \sigma \left(X^{\mathrm{\prime }},X\right)}$-bounded if and only if there exists abarrel${\displaystyle B}$ in${\displaystyle X}$ such that${\displaystyle H\subseteq B^{\circ }.}$^[1]
• Let${\displaystyle (X,Y,b)}$ be apairing and let${\displaystyle \nu }$ be a locally convex topology on${\displaystyle X}$ consistent with duality. Then a subset${\displaystyle B}$ of${\displaystyle X}$ is a barrel in${\displaystyle (X,\nu )}$ if and only if${\displaystyle B}$ is thepolar of some${\displaystyle \sigma (Y,X,b)}$-bounded subset of${\displaystyle Y.}$^[1]
• Suppose${\displaystyle M}$ is a vector subspace of finite codimension in a locally convex space${\displaystyle X}$ and${\displaystyle B\subseteq M.}$ If${\displaystyle B}$ is a barrel (resp.bornivorous barrel, bornivorous disk) in${\displaystyle M}$ then there exists a barrel (resp. bornivorous barrel, bornivorous disk)${\displaystyle C}$ in${\displaystyle X}$ such that${\displaystyle B=C\cap M.}$^[2]

Denote by${\displaystyle L(X;Y)}$ the space of continuous linear maps from${\displaystyle X}$ into${\displaystyle Y.}$

If${\displaystyle (X,\tau )}$ is aHausdorfftopological vector space (TVS) withcontinuous dual space${\displaystyle X^{\mathrm{\prime }}}$ then the following are equivalent:

1. ${\displaystyle X}$ is barrelled.
2. Definition: Every barrel in${\displaystyle X}$ 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${\displaystyle Y}$ with a topology that is not theindiscrete topology is aBaire space if and only if every absorbing balanced subset is a neighborhood ofsome point of${\displaystyle Y}$ (not necessarily the origin).^[2]
3. For any Hausdorff TVS${\displaystyle Y}$ every pointwise bounded subset of${\displaystyle L(X;Y)}$ is equicontinuous.^[3]
4. For anyF-space${\displaystyle Y}$ every pointwise bounded subset of${\displaystyle L(X;Y)}$ is equicontinuous.^[3]
   • An F-space is a completemetrizable TVS.
5. Everyclosed linear operator from${\displaystyle X}$ into a complete metrizable TVS is continuous.^[4]
   • A linear map${\displaystyle F:X\to Y}$ is calledclosed if itsgraph is a closed subset of${\displaystyle X\times Y.}$
6. Every Hausdorff TVS topology${\displaystyle \nu }$ on${\displaystyle X}$ that has a neighborhood basis of the origin consisting of${\displaystyle \tau }$-closed set is course than${\displaystyle \tau .}$^[5]

If${\displaystyle (X,\tau )}$ islocally convex space then this list may be extended by appending:

7. There exists a TVS${\displaystyle Y}$ not carrying theindiscrete topology (so in particular,${\displaystyle Y\ne \{0\}}$) such that every pointwise bounded subset of${\displaystyle L(X;Y)}$ is equicontinuous.^[2]
8. For any locally convex TVS${\displaystyle Y,}$ every pointwise bounded subset of${\displaystyle L(X;Y)}$ is equicontinuous.^[2]
   • 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.
9. Every${\displaystyle \sigma \left(X^{\mathrm{\prime }},X\right)}$-bounded subset of the continuous dual space${\displaystyle X}$ is equicontinuous (this provides a partial converse to theBanach-Steinhaus theorem).^[2][6]
10. ${\displaystyle X}$ carries thestrong dual topology${\displaystyle \beta \left(X,X^{\mathrm{\prime }}\right).}$^[2]
11. Everylower semicontinuousseminorm on${\displaystyle X}$ is continuous.^[2]
12. Every linear map${\displaystyle F:X\to Y}$ into a locally convex space${\displaystyle Y}$ isalmost continuous.^[2]
    • A linear map${\displaystyle F:X\to Y}$ is calledalmost continuous if for every neighborhood${\displaystyle V}$ of the origin in${\displaystyle Y,}$ the closure of${\displaystyle F^{-1}(V)}$ is a neighborhood of the origin in${\displaystyle X.}$
13. Every surjective linear map${\displaystyle F:Y\to X}$ from a locally convex space${\displaystyle Y}$ isalmost open.^[2]
    • This means that for every neighborhood${\displaystyle V}$ of 0 in${\displaystyle Y,}$ the closure of${\displaystyle F(V)}$ is a neighborhood of 0 in${\displaystyle X.}$
14. If${\displaystyle \omega }$ is a locally convex topology on${\displaystyle X}$ such that${\displaystyle (X,\omega )}$ has a neighborhood basis at the origin consisting of${\displaystyle \tau }$-closed sets, then${\displaystyle \omega }$ is weaker than${\displaystyle \tau .}$^[2]

If${\displaystyle X}$ is a Hausdorff locally convex space then this list may be extended by appending:

15. Closed graph theorem: Everyclosed linear operator${\displaystyle F:X\to Y}$ into aBanach space${\displaystyle Y}$ iscontinuous.^[7]
    • The linear operator is calledclosed if itsgraph is a closed subset of${\displaystyle X\times Y.}$
16. For every subset${\displaystyle A}$ of the continuous dual space of${\displaystyle X,}$ the following properties are equivalent:${\displaystyle A}$ is^[6]
    1. equicontinuous;
    2. relatively weakly compact;
    3. strongly bounded;
    4. weakly bounded.
17. The 0-neighborhood bases in${\displaystyle X}$ and the fundamental families of bounded sets in${\displaystyle X_{\beta }^{\mathrm{\prime }}}$ correspond to each other bypolarity.^[6]

If${\displaystyle X}$ ismetrizable topological vector space then this list may be extended by appending:

18. For any complete metrizable TVS${\displaystyle Y}$ every pointwise boundedsequence in${\displaystyle L(X;Y)}$ is equicontinuous.^[3]

If${\displaystyle X}$ is a locally convexmetrizable topological vector space then this list may be extended by appending:

19. (Property S): Theweak* topology on${\displaystyle X^{\mathrm{\prime }}}$ issequentially complete.^[8]
20. (Property C): Every weak* bounded subset of${\displaystyle X^{\mathrm{\prime }}}$ is${\displaystyle \sigma \left(X^{\mathrm{\prime }},X\right)}$-relativelycountably compact.^[8]
21. (𝜎-barrelled): Every countable weak* bounded subset of${\displaystyle X^{\mathrm{\prime }}}$ is equicontinuous.^[8]
22. (Baire-like):${\displaystyle X}$ is not the union of an increase sequence ofnowhere densedisks.^[8]

Each of the following topological vector spaces is barreled:

1. TVSs that areBaire space.
   • Consequently, every topological vector space that is of thesecond category in itself is barrelled.
2. F-spaces,Fréchet spaces,Banach spaces, andHilbert spaces.
   • However, there existnormed vector spaces that arenot barrelled. For example, if the${\displaystyle L^p}$-space${\displaystyle L^2([0,1])}$ is topologized as a subspace of${\displaystyle L^1([0,1]),}$ then it is not barrelled.
3. Completepseudometrizable TVSs.^[9]
   • Consequently, every finite-dimensional TVS is barrelled.
4. Montel spaces.
5. Strong dual spaces of Montel spaces (since they are necessarily Montel spaces).
6. A locally convexquasi-barrelled space that is also aσ-barrelled space.^[10]
7. A sequentially completequasibarrelled space.
8. Aquasi-complete Hausdorff locally convexinfrabarrelled space.^[2]
   • A TVS is calledquasi-complete if every closed and bounded subset is complete.
9. A TVS with a dense barrelled vector subspace.^[2]
   • Thus the completion of a barreled space is barrelled.
10. A Hausdorff locally convex TVS with a denseinfrabarrelled vector subspace.^[2]
    • Thus the completion of an infrabarrelled Hausdorff locally convex space is barrelled.^[2]
11. A vector subspace of a barrelled space that has countable codimensional.^[2]
    • In particular, a finite codimensional vector subspace of a barrelled space is barreled.
12. A locally convex ultrabarelled TVS.^[11]
13. A Hausdorff locally convex TVS${\displaystyle X}$ such that every weakly bounded subset of its continuous dual space is equicontinuous.^[12]
14. A locally convex TVS${\displaystyle X}$ such that for every Banach space${\displaystyle B,}$ a closed linear map of${\displaystyle X}$ into${\displaystyle B}$ is necessarily continuous.^[13]
15. A product of a family of barreled spaces.^[14]
16. A locally convex direct sum and the inductive limit of a family of barrelled spaces.^[15]
17. A quotient of a barrelled space.^[16][15]
18. A Hausdorffsequentially completequasibarrelled boundedly summing TVS.^[17]
19. A locally convex Hausdorffreflexive space is barrelled.

• A barrelled space need not beMontel,complete,metrizable, unordered Baire-like, nor the inductive limit of Banach spaces.
• Not all normed spaces are barrelled. However, they are all infrabarrelled.^[2]
• A closed subspace of a barreled space is not necessarilycountably quasi-barreled (and thus not necessarily barrelled).^[18]
• There exists a dense vector subspace of theFréchet barrelled space${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$ that is not barrelled.^[2]
• There exist complete locally convex TVSs that are not barrelled.^[2]
• Thefinest locally convex topology on an infinite-dimensional vector space is a Hausdorff barrelled space that is ameagre subset of itself (and thus not aBaire space).^[2]

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

TheBanach-Steinhaus theorem is a corollary of the above result.^[20] When the vector space${\displaystyle Y}$ consists of the complex numbers then the following generalization also holds.

Recall that a linear map${\displaystyle F:X\to Y}$ is calledclosed if its graph is a closed subset of${\displaystyle X\times Y.}$

• Every Hausdorff barrelled space isquasi-barrelled.^[23]
• A linear map from a barrelled space into a locally convex space isalmost continuous.
• A linear map from a locally convex spaceonto a barrelled space isalmost open.
• Aseparately continuous bilinear map from a product of barrelled spaces into a locally convex space ishypocontinuous.^[24]
• A linear map with a closed graph from a barreled TVS into a${\displaystyle B_r}$-complete TVS is necessarily continuous.^[13]

• Barrelled set
• Countably barrelled space
• Distinguished space – TVS whose strong dual is barralled
• Quasibarrelled space
• Ultrabarrelled space
• Uniform boundedness principle#Generalisations – Theorem stating that pointwise boundedness implies uniform boundedness
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
• Webbed space – Space where open mapping and closed graph theorems hold

• Voigt, Jürgen (2020).A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Cham:Birkhäuser Basel.ISBN 978-3-030-32945-7.OCLC 1145563701.
• Wilansky, Albert (2013).Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc.ISBN 978-0-486-49353-4.OCLC 849801114.

• Topological vector spaces

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