BOUNDED LINEAR TRANSFORMATIONS BETWEEN PROBABILISTIC NORMED VECTOR-SPACES

We introduce a property of Q-boundedness on linear transformations bet ween probabilistic T-normed vector spaces, where Q and T are lower sem icontinuous triangular norms on the unit interval. We introduce a prop erty of metric Q-continuity on functions between probabilistic T-metri c spaces. We show that Q-boundedness and Q-continuity coincide on the domain of definition of the former property. We show that the already existing fuzzy topological theories fbr probabilistic T-metric spaces correspond either to T-m-continuity or to Min-continuity. By investiga ting the Q-continuity of vector space operations in probabilistic T-no rmed spaces, we find that the condition Q less than or equal to T is a useful one. This indicates that the class of probabilistic T-metric s paces may be the largest possible setting for a rich theory of metric T-continuity; a fuzzy topological accommodation of which is, at presen t, available only when T = T-m and when T = Min.