The aim of this paper is to investigate the nature of bounded sets in a topological ∈-tensor product EX∈* F of any two locally convex topological vector spaces E and F over the same scalar field K. Next, we apply the results of this investigation to the study of each of the following: (a) Totally summable families in EX∈*F; (b) ∈-tensor product of DF-spaces; (c) Topological nature of the dual of E X∈*F, where E and F are strong duals of Banach spaces; (d) Properties of bounded sets in an ∈-tensor product of metrizable spaces.