and for every angle let denote the closed line segment from the origin to the point Let Then is always an absorbing subset of (a real vector space) but it is an absorbing subset of (a complex vector space) if and only if it is a neighborhood of the origin. Moreover, is a balanced subset of if and only if for every (if this is the case then and are completely determined by 's values on ) but is a balanced subset of if and only it is an open or closed ball centered at the origin (of radius ). In particular, barrels in are exactly those closed balls centered at the origin with radius in If then is a closed subset that is absorbing in but not absorbing in and that is neither convex, balanced, nor a neighborhood of the origin in By an appropriate choice of the function it is also possible to have be a balanced and absorbing subset of that is neither closed nor convex. To have be a balanced, absorbing, and closed subset of that is convex nor a neighborhood of the origin, define on as follows: for let (alternatively, it can be any positive function on that is continuously differentiable, which guarantees that and that is closed, and that also satisfies which prevents from being a neighborhood of the origin) and then extend to by defining which guarantees that is balanced in Properties of barrels In any topological vector space (TVS) every barrel in absorbs every compact convex subset of In any locally convex Hausdorff TVS every barrel in absorbs every convex bounded complete subset of If is locally convex then a subset of is -bounded if and only if there exists a barrel in such that Let be a pairing and let be a locally convex topology on consistent with duality. Then a subset of is a barrel in if and only if is the polar of some -bounded subset of Suppose is a vector subspace of finite codimension in a locally convex space and If is a barrel (resp. bornivorous barrel, bornivorous disk) in then there exists a barrel (resp. bornivorous barrel, bornivorous disk) in such that Characterizations of barreled spaces Denote by the space of continuous linear maps from into If is a Hausdorff topological vector space (TVS) with continuous dual space then the following are equivalent: is barrelled. : Every