Some results on barreledness in projective tensor products
Grothendieck showed that if \(E{\hat \otimes}_{\pi}K^{(N)}\) is barrelled, for a Fréchet space E, then E satisfies the following property: (*) every quotient of E with a continuous norm is a Banach space (the so called ''quojection'' by Bellenot and Dubinsky). In this article it is shown that this condition is also sufficient. The following consequences can be obtained: (a) \(L_ b(K^ N,E)\) is barrelled if and only if E satisfies (*) (b) if E is a Mackey df-space, then \(L_ b(E,K^{(N)})\) is barrelled if and only if \(E'_ b\) satisfies (*) (c) if E is a complete space, then the barrelledness of the Schwartz spaces \({\mathcal E}(X,E)\) and \({\mathcal D}(X,E),\) X being an open set of the euclidean space, implies that E is quasi-Baire, i.e., E is barrelled and E cannot be covered by an increasing sequence of proper closed subspaces. The barrelled spaces F for which \(F\otimes_{\pi}K^ N\) is barrelled are characterized as those barrelled spaces which are quasi-Baire, i.e., the barrelled spaces not containing a complemented copy of \(K^{(N)}\).
scientific article; zbMATH DE number 3855785