On extension of continuity and application of countability axioms in topological tensor products

In this paper an attempt has been made to prove an analogue of the result¹ on the extension of linear and continuous mappings in topological vector spaces to the context of topological tensor products. We also prove that the topological tensor-product of two separable locally convex spaces is separable, and the projective tensor product of any two locally-convex 1st countable (2nd countable) spaces is 1st countable (2nd countable). For the terminology, and basic notations, we refer to Treves¹ and Jha.^2,3