On the stability of the completeness of \(L^{\alpha}\) spaces (Q920356)

Let E denote a quasi-complete Hausdorff locally convex space with some topology (\({\mathcal P}\) is a family of seminorms and m is a countable additive measure). The purpose of this paper is the investigation of the stability of the quasi-completeness of the space \(L^{\alpha}(E)\), where \(1\leq \alpha <\infty\). The author considers the very interesting case when \(E=\prod_{n}E_ n\) (Theorem 6) and also the very important case in which \(E=\lim_{\to}E_ n\) (Theorem 3), where \(E_ n\) are the same type as E.