Bounded sets in \({\mathcal L}(E,F)\) (Q1825405)

Let E and F denote Hausdorff locally convex spaces, let B be an absolutely convex subset of E. If the space \(E_ B=\cup \{nB:\) \(n\geq 1\}\) is a barreled normed space, then B is called a barreled disk; E is locally barreled if each bounded set in E is contained in a closed, bounded barreled disk. The main result of this paper is the following characterization of locally barreled spaces in terms of the topologies of the space \({\mathcal L}(E,F)\) of continuous linear maps from E to F: Suppose that for each absolutely convex, closed, bounded set \(A\subset E\) there exists a barrel \(D\subset E\) such that \(A=D\cap E_ A\). Then the following are equivalent: (a) The families of bounded subsets of \({\mathcal L}(E,F)\) are identical for all \({\mathcal S}\)-topologies on \({\mathcal L}(E,F)\), where \({\mathcal S}\) is a family of bounded subsets of E which covers E. (b) E is locally barreled.