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.

Barrels

A convex and balanced subset of a real or complex vector space is called a and it is said to be, , or. A **' or a **' 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 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 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 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 X is equal to \Complex (if considered as a complex vector space) or equal to \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 be any function and for every angle let S_{\theta} denote the closed line segment from the origin to the point Let Then S is always an absorbing subset of \R^2 (a real vector space) but it is an absorbing subset of \Complex (a complex vector space) if and only if it is a neighborhood of the origin. Moreover, S is a balanced subset of \R^2 if and only if for every (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 \Complex if and only it is an open or closed ball centered at the origin (of radius ). In particular, barrels in \Complex are exactly those closed balls centered at the origin with radius in If then S is a closed subset that is absorbing in \R^2 but not absorbing in \Complex, 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 \R^2 that is neither closed nor convex. To have S be a balanced, absorbing, and closed subset of \R^2 that is convex nor a neighborhood of the origin, define R on [0, \pi) as follows: for let (alternatively, it can be any positive function on [0, \pi) that is continuously differentiable, which guarantees that and that S is closed, and that also satisfies which prevents S from being a neighborhood of the origin) and then extend R to by defining which guarantees that S is balanced in \R^2.

Properties of barrels

<ul> <li>In any [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) X, every barrel in X [absorbs](https://bliptext.com/articles/absorbing-set) every compact convex subset of X.</li> <li>In any [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) Hausdorff TVS X, every barrel in X absorbs every convex bounded complete subset of X.</li> <li>If X is locally convex then a subset H of X^{\prime} is -bounded if and only if there exists a [barrel](https://bliptext.com/articles/) B in X such that </li> <li>Let (X, Y, b) be a [pairing](https://bliptext.com/articles/dual-system) 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](https://bliptext.com/articles/polar-set) of some -bounded subset of Y.</li> <li>Suppose M is a vector subspace of finite codimension in a locally convex space X and If B is a barrel (resp. [bornivorous](https://bliptext.com/articles/bornivorous-set) barrel, [bornivorous](https://bliptext.com/articles/bornivorous-set) disk) in M then there exists a barrel (resp. [bornivorous](https://bliptext.com/articles/bornivorous-set) barrel, [bornivorous](https://bliptext.com/articles/bornivorous-set) disk) C in X such that </li> </ul>

Characterizations of barreled spaces

Denote by L(X; Y) the space of continuous linear maps from X into Y. If (X, \tau) is a Hausdorff topological vector space (TVS) with continuous dual space X^{\prime} then the following are equivalent:

<ol> <li>X is barrelled.</li> ****<****l****i****>****:**** ****E****v****e****r****y**** ****b****a****r****r****e****l**** ****i****n**** ****X**** ****i****s**** ****a**** ****n****e****i****g****h****b****o****r****h****o****o****d**** ****o****f**** ****t****h****e**** ****o****r****i****g****i****n****.**** <li>For any Hausdorff TVS Y every pointwise bounded subset of L(X; Y) is equicontinuous.</li> <li>For any [F-space](https://bliptext.com/articles/f-space) Y every pointwise bounded subset of L(X; Y) is equicontinuous. <li>Every [closed linear operator](https://bliptext.com/articles/closed-linear-operator) from X into a complete metrizable TVS is continuous. <li>Every Hausdorff TVS topology \nu on X that has a neighborhood basis of the origin consisting of \tau-closed set is course than \tau.</li> </ol> If (X, \tau) is [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) space then this list may be extended by appending: <li>There exists a TVS Y not carrying the [indiscrete topology](https://bliptext.com/articles/indiscrete-topology) (so in particular, ) such that every pointwise bounded subset of L(X; Y) is equicontinuous.</li> <li>For any locally convex TVS Y, every pointwise bounded subset of L(X; Y) is equicontinuous. <li>Every -bounded subset of the continuous dual space X is equicontinuous (this provides a partial converse to the [Banach-Steinhaus theorem](https://bliptext.com/articles/banach-steinhaus-theorem)). </li> <li>X carries the [strong dual topology](https://bliptext.com/articles/strong-dual-topology) </li> <li>Every [lower semicontinuous](https://bliptext.com/articles/lower-semicontinuous) [seminorm](https://bliptext.com/articles/seminorm) on X is continuous.</li> <li>Every linear map F : X \to Y into a locally convex space Y is [almost continuous](https://bliptext.com/articles/almost-continuous). <li>Every surjective linear map F : Y \to X from a locally convex space Y is [almost open](https://bliptext.com/articles/almost-open-map). <li>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.</li> </ol> If X is a Hausdorff locally convex space then this list may be extended by appending: <li>**[Closed graph theorem](https://bliptext.com/articles/closed-graph-theorem)**: Every [closed linear operator](https://bliptext.com/articles/closed-linear-operator) F : X \to Y into a [Banach space](https://bliptext.com/articles/banach-space) Y is [continuous](https://bliptext.com/articles/continuous-linear-operator). <li>For every subset A of the continuous dual space of X, the following properties are equivalent: A is <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 X and the fundamental families of bounded sets in correspond to each other by [polarity](https://bliptext.com/articles/polar-set). </li> </ol> If X is [metrizable topological vector space](https://bliptext.com/articles/metrizable-topological-vector-space) then this list may be extended by appending: <li>For any complete metrizable TVS Y every pointwise bounded in L(X; Y) is equicontinuous.</li> </ol> If X is a locally convex [metrizable topological vector space](https://bliptext.com/articles/metrizable-topological-vector-space) then this list may be extended by appending: <li>: The [weak* topology](https://bliptext.com/articles/weak-topology) on X^{\prime} is [sequentially complete](https://bliptext.com/articles/sequentially-complete).</li> <li>: Every weak* bounded subset of X^{\prime} is -relatively [countably compact](https://bliptext.com/articles/countably-compact).</li> <li>: Every countable weak* bounded subset of X^{\prime} is equicontinuous.</li> <li>: X is not the union of an increase sequence of [nowhere dense](https://bliptext.com/articles/nowhere-dense-set) [disks](https://bliptext.com/articles/absolutely-convex-set).</li> </ol>

Examples and sufficient conditions

Each of the following topological vector spaces is barreled:

<ol> <li>TVSs that are [Baire space](https://bliptext.com/articles/baire-space). <li>[F-spaces](https://bliptext.com/articles/f-space), [Fréchet spaces](https://bliptext.com/articles/fr-chet-space), [Banach spaces](https://bliptext.com/articles/banach-space), and [Hilbert spaces](https://bliptext.com/articles/hilbert-space). <li>[Complete](https://bliptext.com/articles/complete-topological-vector-space) [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) TVSs. <li>[Montel spaces](https://bliptext.com/articles/montel-space).</li> <li>[Strong dual spaces](https://bliptext.com/articles/strong-dual-space) of Montel spaces (since they are necessarily Montel spaces).</li> <li>A locally convex [quasi-barrelled space](https://bliptext.com/articles/quasi-barrelled-space) that is also a [σ-barrelled space](https://bliptext.com/articles/barrelled-space).</li> <li>A sequentially complete [quasibarrelled space](https://bliptext.com/articles/quasibarrelled-space).</li> <li>A [quasi-complete](https://bliptext.com/articles/quasi-complete-space) Hausdorff locally convex [infrabarrelled space](https://bliptext.com/articles/infrabarrelled-space). <li>A TVS with a dense barrelled vector subspace. <li>A Hausdorff locally convex TVS with a dense [infrabarrelled](https://bliptext.com/articles/infrabarrelled-space) vector subspace. <li>A vector subspace of a barrelled space that has countable codimensional. <li>A locally convex ultrabarelled TVS.</li> <li>A Hausdorff locally convex TVS X such that every weakly bounded subset of its continuous dual space is equicontinuous.</li> <li>A locally convex TVS X such that for every Banach space B, a closed linear map of X into B is necessarily continuous.</li> <li>A product of a family of barreled spaces.</li> <li>A locally convex direct sum and the inductive limit of a family of barrelled spaces.</li> <li>A quotient of a barrelled space.</li> <li>A Hausdorff [sequentially complete](https://bliptext.com/articles/sequentially-complete) [quasibarrelled](https://bliptext.com/articles/quasibarrelled) boundedly summing TVS.</li> <li>A locally convex Hausdorff [reflexive space](https://bliptext.com/articles/reflexive-space) is barrelled.</li> </ol>

Counter examples

<ul> <li>A barrelled space need not be [Montel](https://bliptext.com/articles/montel-space), [complete](https://bliptext.com/articles/complete-topological-vector-space), [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space), unordered Baire-like, nor the inductive limit of Banach spaces.</li> <li>Not all normed spaces are barrelled. However, they are all infrabarrelled.</li> <li>A closed subspace of a barreled space is not necessarily [countably quasi-barreled](https://bliptext.com/articles/countably-quasi-barrelled-space) (and thus not necessarily barrelled).</li> <li>There exists a dense vector subspace of the [Fréchet](https://bliptext.com/articles/fr-chet-space) barrelled space \R^{\N} that is not barrelled.</li> <li>There exist complete locally convex TVSs that are not barrelled.</li> <li>The [finest locally convex topology](https://bliptext.com/articles/locally-convex-topological-vector-space) on an infinite-dimensional vector space is a Hausdorff barrelled space that is a [meagre](https://bliptext.com/articles/meager-set) subset of itself (and thus not a [Baire space](https://bliptext.com/articles/baire-space)).</li> </ul>

Properties of barreled spaces

Banach–Steinhaus generalization

The importance of barrelled spaces is due mainly to the following results. The Banach-Steinhaus theorem is a corollary of the above result. When the vector space Y consists of the complex numbers then the following generalization also holds. Recall that a linear map F : X \to Y is called closed if its graph is a closed subset of X \times Y.

Other properties

<ul> <li>Every Hausdorff barrelled space is [quasi-barrelled](https://bliptext.com/articles/quasi-barrelled).</li> <li>A linear map from a barrelled space into a locally convex space is [almost continuous](https://bliptext.com/articles/).</li> <li>A linear map from a locally convex space to a barrelled space is [almost open](https://bliptext.com/articles/almost-open-map).</li> <li>A [separately continuous](https://bliptext.com/articles/separately-continuous) bilinear map from a product of barrelled spaces into a locally convex space is [hypocontinuous](https://bliptext.com/articles/hypocontinuous).</li> <li>A linear map with a closed graph from a barreled TVS into a B_r-complete TVS is necessarily continuous.</li> </ul>