• Dr. Upendra Kumar Nirala

The 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 two Baire spaces is identical with the dual space of the π- tensor product to the same two spaces and is also identical with the space of continuous bilinear forms defined on the cross product of the same two Baire spaces. . 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.

Tensor Products, Dual space, ε– tensor product, π- tensor product, bilinear forms)

" Contribution to the Duality theory of tensor products of Baire Space", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.6, Issue 6, page no.710-715, June 2019, Available :http://www.jetir.org/papers/JETIR1906J40.pdf

" Contribution to the Duality theory of tensor products of Baire Space", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.6, Issue 6, page no. pp710-715, June 2019, Available at : http://www.jetir.org/papers/JETIR1906J40.pdf