On linear topologies determined by a family of subsets of a topological vector space

We provide a general framework for the study of the finest linear (locally convex) topology which coincides on a family of subsets with a given linear (locally convex) topology. It is proved that the formation of such topologies always commutes with linear direct sums. We characterize the corresponding situation for products and prove a result about locally convex direct sums sufficiently general to cover the examples which already occurred in the literature. Moreover the 0-nbhd. filters of such topologies are characterized, and several examples are considered.