A convex and balanced subset of a real or complex vector space is called a disk and it is said to be disked, absolutely convex, or convex balanced.

A barrel or a 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 ${\displaystyle \dim X\geq 2}$  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 a closed subset of ${\displaystyle X}$  is the only defining property that does not depend solely on ${\displaystyle 2}$  (or lower)-dimensional vector subspaces of ${\displaystyle X.}$ 

If ${\displaystyle X}$  is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in ${\displaystyle X}$  (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 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 edit

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 ${\displaystyle R:[0,2\pi )\to (0,\infty ]}$  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 ${\textstyle S:=\bigcup _{\theta \in [0,2\pi )}S_{\theta }.}$  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 a neighborhood 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 ${\displaystyle 0\leq \theta <\pi }$  (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 ${\displaystyle 0<r\leq \infty }$ ). In particular, barrels in ${\displaystyle \mathbb {C} }$  are exactly those closed balls centered at the origin with radius in ${\displaystyle (0,\infty ].}$  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 is neither convex nor a neighborhood of the origin, define ${\displaystyle R}$  on ${\displaystyle [0,\pi )}$  as follows: for ${\displaystyle 0\leq \theta <\pi ,}$  let ${\displaystyle R(\theta ):=\pi -\theta }$  (alternatively, it can be any positive function on ${\displaystyle [0,\pi )}$  that is continuously differentiable, which guarantees that ${\textstyle \lim _{\theta \searrow 0}R(\theta )=R(0)>0}$  and that ${\displaystyle S}$  is closed, and that also satisfies ${\textstyle \lim _{\theta \nearrow \pi }R(\theta )=0,}$  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 ${\displaystyle \mathbb {R} ^{2}.}$ 

Properties of barrels edit

• In any topological vector space (TVS) ${\displaystyle X,}$  every barrel in ${\displaystyle X}$  absorbs every compact convex subset of ${\displaystyle X.}$ ^[1]
• In any locally 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 ${\displaystyle X^{\prime }}$  is ${\displaystyle \sigma \left(X^{\prime },X\right)}$ -bounded if and only if there exists a barrel ${\displaystyle B}$  in ${\displaystyle X}$  such that ${\displaystyle H\subseteq B^{\circ }.}$ ^[1]
• Let ${\displaystyle (X,Y,b)}$  be a pairing 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 the polar 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 a Hausdorff topological vector space (TVS) with continuous dual space ${\displaystyle X^{\prime }}$  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 the indiscrete topology is a Baire space if and only if every absorbing balanced subset is a neighborhood of some 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 any F-space ${\displaystyle Y}$  every pointwise bounded subset of ${\displaystyle L(X;Y)}$  is equicontinuous.^[3]
   • An F-space is a complete metrizable TVS.
5. Every closed linear operator from ${\displaystyle X}$  into a complete metrizable TVS is continuous.^[4]
   • A linear map ${\displaystyle F:X\to Y}$  is called closed if its graph 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 ${\displaystyle (X,\tau )}$  is locally convex space then this list may be extended by appending:

7. There exists a TVS ${\displaystyle Y}$  not carrying the indiscrete topology (so in particular, ${\displaystyle Y\neq \{0\}}$ ) 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 principal holds.
9. Every ${\displaystyle \sigma \left(X^{\prime },X\right)}$ -bounded subset of the continuous dual space ${\displaystyle X}$  is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem).^[2][6]
10. ${\displaystyle X}$  carries the strong dual topology ${\displaystyle \beta \left(X,X^{\prime }\right).}$ ^[2]
11. Every lower semicontinuous seminorm on ${\displaystyle X}$  is continuous.^[2]
12. Every linear map ${\displaystyle F:X\to Y}$  into a locally convex space ${\displaystyle Y}$  is almost continuous.^[2]
    • A linear map ${\displaystyle F:X\to Y}$  is called almost 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 ${\displaystyle Y}$  is almost 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: Every closed linear operator ${\displaystyle F:X\to Y}$  into a Banach space ${\displaystyle Y}$  is continuous.^[7]
    • The linear operator is called closed if its graph 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 ${\displaystyle X_{\beta }^{\prime }}$  correspond to each other by polarity.^[6]

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

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

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

19. (Property S): The weak* topology on ${\displaystyle X^{\prime }}$  is sequentially complete.^[8]
20. (Property C): Every weak* bounded subset of ${\displaystyle X^{\prime }}$  is ${\displaystyle \sigma \left(X^{\prime },X\right)}$ -relatively countably compact.^[8]
21. (𝜎-barrelled): Every countable weak* bounded subset of ${\displaystyle X^{\prime }}$  is equicontinuous.^[8]
22. (Baire-like): ${\displaystyle X}$  is not the union of an increase sequence of nowhere dense disks.^[8]

Each of the following topological vector spaces is barreled:

1. TVSs that are Baire space.
   • Consequently, every topological vector space that is of the second category in itself is barrelled.
2. F-spaces, Fréchet spaces, Banach spaces, and Hilbert spaces.
   • However, there exist normed vector spaces that are not 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. Complete pseudometrizable 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 convex quasi-barrelled space that is also a σ-barrelled space.^[10]
7. A sequentially complete quasibarrelled space.
8. A quasi-complete Hausdorff locally convex infrabarrelled space.^[2]
   • A TVS is called quasi-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 dense infrabarrelled 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 Hausdorff sequentially complete quasibarrelled boundedly summing TVS.^[17]
19. A locally convex Hausdorff reflexive space is barrelled.

Counter examples edit

• A barrelled space need not be Montel, 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 necessarily countably quasi-barreled (and thus not necessarily barrelled).^[18]
• There exists a dense vector subspace of the Fréchet barrelled space ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$  that is not barrelled.^[2]
• There exist complete locally convex TVSs that are not barrelled.^[2]
• 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).^[2]

Banach–Steinhaus generalization edit

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

The Banach-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 called closed if its graph is a closed subset of ${\displaystyle X\times Y.}$ 

Other properties edit

• Every Hausdorff barrelled space is quasi-barrelled.^[23]
• A linear map from a barrelled space into a locally convex space is almost continuous.
• A linear map from a locally convex space onto a barrelled space is almost open.
• A separately continuous bilinear map from a product of barrelled spaces into a locally convex space is hypocontinuous.^[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 – A 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.