On the Existence of a Norm Weaker than a Given Family of Norms on a Vector Space

Abstract

Let N_i(x) be a sequence of norms defined over a vector space E. Let us consider all those linear topologies T over E which are weaker than N_i(x) for all i. (For brevity N_i(x) is used to denote the norm as well as the corresponding topology). These include the lattice product tolopogy [3] of the N_i(x), i.e. the strongest topology weaker than each of the N_i(x) topologies.