Quojections and projective tensor products (Q762410)

This article is devoted to the study of stability of barrelledness properties of projective tensor products of a Fréchet space E and a barrelled space F. The following results are proved: (a) Let E be a Fréchet space. The following are equivalent: 1. Every quotient of E with a continuous norm is Banach space. (These Fréchet spaces are called quojections by Bellenot and Dubinsky). 2. E\({\hat \otimes}_{\pi}F\) is barrelled for every barrelled space F. (b) Let E be a reflexive Fréchet space. The following are equivalent: 1. E is a quojection. 2. \(E\otimes_{\pi}F\) is barrelled for every barrelled space F which does not contain \(K^{(N)}\) complemented. 3. \(E{\hat \otimes}_{\pi}\lambda (A)'\!_ b\) is barrelled for every nuclear Köthe space \(\lambda\) (A) with a continuous norm. Result (b) is obtained as a consequence of a characterization without the assumption of reflexivity.