On sufficient conditions for the sum of two weak closed convex sets to be weak closed

Abstract

1. The question as to when the sum of two closed sets is closed arises naturally in applied mathematics and is of interest in its own right. For subsets of an infinite dimensional space, answers to this question have been given by Choquet [1], Ky Fan [3], Dieudonn+ [2] and more recently by Jameson [4]. Ky Fan, in particular, has shown that the sum of two weakly closed convex subsets of a locally convex topological vector space is weakly closed under a condition that takes account of the mutual position of the two sets. In this note, we offer another sufficient condition of this genre for subsets in the dual of a locally convex topological vector space. Despite its simplicity, this result seems to have been overlooked in the literature. We also give simple examples which bring out the difference between our sufficient condition and those of the others. It is worth mentioning that our result is directly inspired by a problem in mathematical economies, see [5]. 2. Let (E, z) be a real, Hausdorff locally convex vector space and E' its topological dual, i.e., the space of all z-continuous linear functions on E. a (E', E) denotes the weak �9 topology on E'. Following [6, Definition 12-3-7], we say that E is strictly hypercomplete if for any S c E', S n U ~ is weak �9 compact implies S is weak �9 closed, where U ~ is the polar of any neighborhood of zero in E. We can now state