• Dr. Upendra Kumar Nirala

Abstract :- The Basic and authentic study of a locally convex space in terms of its dual is the central part of the modern theory of topological vector spaces, for it provides the setting for the deepest and most beautiful results of the subject. Various authors have mentioned In this paper we have proved that the dual space of the ε – tensor product of the two given metrizable locally convex spaces is equal to the ε – tensor product of their topological dual spaces . Duality theory of ε – tensor product of two metrizable locally convex spaces considered about topological dual space of tensor products. objective of this paper is For all our purposes, topological vector spaces are locally convex, in the sense of having a basis at consisting of convex opens. We prove below that a separating family of seminorms produces a locally convex topology. Conversely, every locally convex topology is given by separating families of semi-norms: the seminorms are functionals associated to a local basis of balanced, convex opens. Giving the topology on a locally convex V by a family of seminorms exhibits V as a dense subspace of a projective limit of Banach spaces, with the subspace topology. This chapter presents the most basic results on topological vector spaces. With the exception of the last section, the scalar field over which vector spaces are defined can be an arbitrary.

Keywords :- ε – tensor product, Metrizable, Locally, convex spaces, Topological dual spaces

" contribution to the duality theory of tensor products of two metrizable locally convex spaces is equal to product of their topological dual spaces ", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.7, Issue 10, page no.1769-1771, October-2020, Available :http://www.jetir.org/papers/JETIR2010229.pdf

" contribution to the duality theory of tensor products of two metrizable locally convex spaces is equal to product of their topological dual spaces ", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.7, Issue 10, page no. pp1769-1771, October-2020, Available at : http://www.jetir.org/papers/JETIR2010229.pdf