Contribution of the Duality Theory of epsilon - And Pi-Tensor Products of Baire Spaces

The basic study of the tensor product of the duality theory is 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 semi norms produces a locally convex topology. Conversely, every locally convex topology is given by separating families of semi-norms: the semi norms are functionals associated to a local basis of balanced, convex opens. Giving the topology on a locally convex V by a family of semi norms 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