Barrelled space

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 $\dim X \geq 2$ 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 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 R : , 2\pi) \to (0, \infty/math> be any function and for every angle $\theta \in [0, 2 \pi),$ let $S_$ denote the closed line segment from the origin to the point $R(\theta) e^ \in \Complex.$ Let $S := \bigcup_ S_.$ 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 $R(\theta) = R(\pi + \theta)$ 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 $\Complex$ if and only it is an open or closed ball centered at the origin (of radius $0 < r \leq \infty$). In particular, barrels in $\Complex$ 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 $\R^2$ 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 \lim__R(\theta)_=_R(0)_>_0_and_that_$S$_is_closed,_and_that_also_satisfies_\lim__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_$\R^2.$

_Properties_of_barrels

• In_any_topological_vector_space_

  In_mathematics,_a_topological_vector_space_(also_called_a_linear_topological_space_and_commonly_abbreviated_TVS_or_t.v.s.)_is_one_of_the_basic_structures_investigated_in_functional_analysis.
  A_topological_vector_space_is_a_vector_space_that_is_als_...