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 ε>0 there exists an x∈ B so that x co (B T-1(T(x)+ε V)) (respectively, so that x s-co (B T-1(T(x)+ε V)), respectively, so that x co (B T-1(T(x)+ε V)) ). Moreover, B is called T-dentable (respectively, T-s-dentable, T-f-dentable) if it is T-V-dentable (respectively, T-V-s-dentable, T-V-f-dentable) 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: 1) T ∈ RN(E,F); 2) Each bounded set in E is T-dentable; 3) Each bounded set in E is T-s-dentable; 4) Each bounded set in E is T-f-dentable. Therefore, we have a generalization of Theorem 1 in [8], which gave a geometric characterization of RN-operators between Banach spaces.