Topological Direct Sums of non Convex Barrelled Spaces

Using the concept of the strings in the vector spaces, is developed a theory relatedto the topological vector spaces (t.v.s).At this point of view, there are some important definitions for the strings and wecan also see their characteristics in the topological vector spaces.Also, considering a set of t.v.s we show that the topological product and thetopological direct sum coincide if and only if I is finite. We want to show somepermanence properties of barrelled spaces and conclude, every t.v.s of second category(i.e a Baire space) is barrelled. Especially (F)-spaces are barrelled. Some important resultsare: The topological direct sum (the product) of barrelled spaces is barrelled. Everyquotient space of a barrelled space is barrelled. Finally, we will show that is also true for asubspace F of finite codimension in the barrelled space E the topology induced on F by Eis barrelled