Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Abstract

For locally convex vector spaces (l.c.v.s.) E and F and for linear and continuous operator T : E → F and for an absolutely convex neighborhood V of zero in F , a bounded subset B of E is said to be T -V-dentable (respectively, T -V-s-dentable, respectively, T -V-f-dentable ) if for any ǫ > 0 there exists an x ∈ B so that) for every absolutely convex neighborhood V of zero in F.RN-operators between locally convex vector spaces have been introduced in [5]. We present a theorem which says that, for a large class of l.c.v.s. E, F, if T : E → F is a linear continuous map, then the following are equivalent:Therefore, we have a generalization of Theorem 1 in [8], which gave a geometric characterization of RN-operators between Banach spaces.