Intersections of Fréchet Schwartz spaces and their duals

Abstract

In this article the structure of the intersections of a Frechet Schwartz space F and a (DFS)-space E=ind n E n is investigated. A complete characterization of the locally convex properties of E ⋃ F is given. This space is boraological if and only if the inductive limit E + F is complete. The results are based on recent progress on the structure of (LF)-spaces. The article includes examples of (FS)-spaces F and (DFS)-spaces E such that there are sequentially continuous linear forms on E ⋃ F which are not continuous, thus answering a question of Langenbruch.