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 <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 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 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 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></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></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></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></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></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>: 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 point of <math>Y</math> (not necessarily the origin).</li>

<li>For any Hausdorff TVS <math>Y</math> every pointwise bounded subset of <math>L(X; Y)</math> is equicontinuous.</li>

<li>For any F-space <math>Y</math> every pointwise bounded subset of <math>L(X; Y)</math> is equicontinuous.

  • 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.

  • 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></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.</li>

<li>For any locally convex TVS <math>Y,</math> every pointwise bounded subset of <math>L(X; Y)</math> is equicontinuous.

  • 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).</li>

<li><math>X</math> carries the strong dual topology <math>\beta\left(X, X^{\prime}\right).</math></li>

<li>Every lower semicontinuous seminorm on <math>X</math> is continuous.</li>

<li>Every linear map <math>F : X \to Y</math> into a locally convex space <math>Y</math> is almost continuous.

  • A linear map <math>F : X \to Y</math> is called 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.

  • 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></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.

  • The linear operator is called 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