Every barrel must contain the origin. If $\dim X\geq 2$ and if $S$ is any subset of $X,$ then $S$ is a convex, balanced, and absorbing set of $X$ if and only if this is all true of $S\cap Y$ in $Y$ for every $2$-dimensional vector subspace $Y;$ thus if $\dim X>2$ then the requirement that a barrel be a closed subset of $X$ is the only defining property that does not depend solely on $2$ (or lower)-dimensional vector subspaces of $X.$
If $X$ is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in $X$ (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.
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 $X$ is equal to $\mathbb {C}$ (if considered as a complex vector space) or equal to $\mathbb {R} ^{2))$ (if considered as a real vector space). Regardless of whether $X$ is a real or complex vector space, every barrel in $X$ is necessarily a neighborhood of the origin (so $X$ is an example of a barrelled space). Let $R:[0,2\pi )\to (0,\infty ]$ be any function and for every angle $\theta \in [0,2\pi ),$ let $S_{\theta ))$ denote the closed line segment from the origin to the point $R(\theta )e^{i\theta }\in \mathbb {C} .$ Let ${\textstyle S:=\bigcup _{\theta \in [0,2\pi )}S_{\theta }.}$ Then $S$ is always an absorbing subset of $\mathbb {R} ^{2))$ (a real vector space) but it is an absorbing subset of $\mathbb {C}$ (a complex vector space) if and only if it is a neighborhood of the origin. Moreover, $S$ is a balanced subset of $\mathbb {R} ^{2))$ if and only if $R(\theta )=R(\pi +\theta )$ for every $0\leq \theta <\pi$ (if this is the case then $R$ and $S$ are completely determined by $R$'s values on $[0,\pi )$) but $S$ is a balanced subset of $\mathbb {C}$ if and only it is an open or closed ball centered at the origin (of radius $0<r\leq \infty$). In particular, barrels in $\mathbb {C}$ are exactly those closed balls centered at the origin with radius in $(0,\infty ].$ If $R(\theta ):=2\pi -\theta$ then $S$ is a closed subset that is absorbing in $\mathbb {R} ^{2))$ but not absorbing in $\mathbb {C} ,$ and that is neither convex, balanced, nor a neighborhood of the origin in $X.$ By an appropriate choice of the function $R,$ it is also possible to have $S$ be a balanced and absorbing subset of $\mathbb {R} ^{2))$ that is neither closed nor convex. To have $S$ be a balanced, absorbing, and closed subset of $\mathbb {R} ^{2))$ that is neither convex nor a neighborhood of the origin, define $R$ on $[0,\pi )$ as follows: for $0\leq \theta <\pi ,$ let $R(\theta ):=\pi -\theta$ (alternatively, it can be any positive function on $[0,\pi )$ that is continuously differentiable, which guarantees that ${\textstyle \lim _{\theta \searrow 0}R(\theta )=R(0)>0}$ and that $S$ is closed, and that also satisfies ${\textstyle \lim _{\theta \nearrow \pi }R(\theta )=0,}$ which prevents $S$ from being a neighborhood of the origin) and then extend $R$ to $[\pi ,2\pi )$ by defining $R(\theta ):=R(\theta -\pi ),$ which guarantees that $S$ is balanced in $\mathbb {R} ^{2}.$
In any locally convex Hausdorff TVS $X,$ every barrel in $X$ absorbs every convex bounded complete subset of $X.$^{[1]}
If $X$ is locally convex then a subset $H$ of $X^{\prime ))$ is $\sigma \left(X^{\prime },X\right)$-bounded if and only if there exists a barrel$B$ in $X$ such that $H\subseteq B^{\circ }.$^{[1]}
Let $(X,Y,b)$ be a pairing and let $\nu$ be a locally convex topology on $X$ consistent with duality. Then a subset $B$ of $X$ is a barrel in $(X,\nu )$ if and only if $B$ is the polar of some $\sigma (Y,X,b)$-bounded subset of $Y.$^{[1]}
Suppose $M$ is a vector subspace of finite codimension in a locally convex space $X$ and $B\subseteq M.$ If $B$ is a barrel (resp. bornivorous barrel, bornivorous disk) in $M$ then there exists a barrel (resp. bornivorous barrel, bornivorous disk) $C$ in $X$ such that $B=C\cap M.$^{[2]}
Definition: Every barrel in $X$ 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 $Y$ 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 $Y$ (not necessarily the origin).^{[2]}
For any Hausdorff TVS $Y$ every pointwise bounded subset of $L(X;Y)$ is equicontinuous.^{[3]}
For any F-space$Y$ every pointwise bounded subset of $L(X;Y)$ is equicontinuous.^{[3]}
Every closed linear operator from $X$ into a complete metrizable TVS is continuous.^{[4]}
A linear map $F:X\to Y$ is called closed if its graph is a closed subset of $X\times Y.$
Every Hausdorff TVS topology $\nu$ on $X$ that has a neighborhood basis of the origin consisting of $\tau$-closed set is course than $\tau .$^{[5]}
If $(X,\tau )$ is locally convex space then this list may be extended by appending:
There exists a TVS $Y$ not carrying the indiscrete topology (so in particular, $Y\neq \{0\))$) such that every pointwise bounded subset of $L(X;Y)$ is equicontinuous.^{[2]}
For any locally convex TVS $Y,$ every pointwise bounded subset of $L(X;Y)$ 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.
Every $\sigma \left(X^{\prime },X\right)$-bounded subset of the continuous dual space $X$ is equicontinuous (this provides a partial converse to the Banach-Steinhaus theorem).^{[2]}^{[6]}
Every linear map $F:X\to Y$ into a locally convex space $Y$ is almost continuous.^{[2]}
A linear map $F:X\to Y$ is called almost continuous if for every neighborhood $V$ of the origin in $Y,$ the closure of $F^{-1}(V)$ is a neighborhood of the origin in $X.$
Every surjective linear map $F:Y\to X$ from a locally convex space $Y$ is almost open.^{[2]}
This means that for every neighborhood $V$ of 0 in $Y,$ the closure of $F(V)$ is a neighborhood of 0 in $X.$
If $\omega$ is a locally convex topology on $X$ such that $(X,\omega )$ has a neighborhood basis at the origin consisting of $\tau$-closed sets, then $\omega$ is weaker than $\tau .$^{[2]}
If $X$ is a Hausdorff locally convex space then this list may be extended by appending:
However, there exist normed vector spaces that are not barrelled. For example, if the $L^{p))$-space$L^{2}([0,1])$ is topologized as a subspace of $L^{1}([0,1]),$ then it is not barrelled.
The importance of barrelled spaces is due mainly to the following results.
Theorem^{[19]} — Let $X$ be a barrelled TVS and $Y$ be a locally convex TVS.
Let $H$ be a subset of the space $L(X;Y)$ of continuous linear maps from $X$ into $Y$.
The following are equivalent:
$H$ is bounded for the topology of pointwise convergence;
$H$ is bounded for the topology of bounded convergence;
The Banach-Steinhaus theorem is a corollary of the above result.^{[20]} When the vector space $Y$ consists of the complex numbers then the following generalization also holds.
Theorem^{[21]} — If $X$ is a barrelled TVS over the complex numbers and $H$ is a subset of the continuous dual space of $X$, then the following are equivalent:
$H$ is weakly bounded;
$H$ is strongly bounded;
$H$ is equicontinuous;
$H$ is relatively compact in the weak dual topology.
Recall that a linear map $F:X\to Y$ is called closed if its graph is a closed subset of $X\times Y.$
Closed Graph Theorem^{[22]} — Every closed linear operator from a Hausdorff barrelled TVS into a complete metrizable TVS is continuous.
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN978-3-540-08662-8. OCLC297140003.
Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN978-0-387-90081-0. OCLC878109401.
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN978-3-642-64988-2. MR0248498. OCLC840293704.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN978-1584888666. OCLC144216834.
Osborne, Mason Scott (2013). Locally Convex Spaces. Graduate Texts in Mathematics. Vol. 269. Cham Heidelberg New York Dordrecht London: Springer Science & Business Media. ISBN978-3-319-02045-7. OCLC865578438.
Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 65–75.
Voigt, Jürgen (2020). A Course on Topological Vector Spaces. Compact Textbooks in Mathematics. Cham: Birkhäuser Basel. ISBN978-3-030-32945-7. OCLC1145563701.