Minimal-type locally convex spaces

Abstract

According to ([1], Chapter IV, § 1, Ex. 13), a real or complex locally convex Hausdorff topological vector (abbreviated to 1.c.) space E~ is called minimal ff there exists no locally convex Hausdorff topology v on E which is strictly coarser than u. The purpose of this paper is to extend this notion of minimality. The 1.c. spaces satisfying the condition of minimali ty in the sense to be made precise in the sequel will be called minimal-type spaces. We shall show tha t the closed graph and open mapping theorems for minimal-type spaces ([1], Chapter IV, § 1, Ex. 13, 14) can be derived from more general theorems. Actually we shall show tha t minimal-type spaces constitute a proper subclass of a class of l.c. spaces for which the above-mentioned theorems are known to be true. § 2. Notations and definitions