Barrelledness and bornological conditions on spaces of vector-valued \(\mu\)-simple functions
The space \(S(\mu, E)\) of classes of \(\mu\)-a.e.-equal simple functions defined on a non-trivial measure space with values in a locally convex space \(E\) is considered with the topology of uniform convergence. It is shown that \(S(\mu, E)\) is quasi-barrelled (resp. bornological) if and only if \(E\) is of the same type and \(E_ \beta^ \prime \) has property \((B)\) of Pietsch. Furthermore \(S(\mu, E)\) is barrelled (resp. DF) if and only if \(E\) is barrelled and nuclear (resp. \(E\) is DF). The space \(S(\mu, E)\) is never ultrabornological. The proofs are based on the isomorphism \(E(\mu, E)\cong S(\mu, \mathbb{K})\otimes_ \varepsilon E\) and on Stone's representation theorem.
scientific article; zbMATH DE number 62169
scientific article