Document Type: Original Manuscript


Department of mathematical sciences, Shahrood university of technology, P.O.Box 3619995161-316, Shahrood, Iran.


We study the theory of best approximation in tensor product and the direct sum of some lattice normed spaces
X_{i}. We introduce quasi tensor product space and discuss about the relation between tensor product space and this
new space which we denote it by X boxtimes Y. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downward or upward and we call them I_{m}-quasi downward or I_{m}-quasi upward.We show that these sets can be interpreted as downward or upward sets. The relation of these sets with
downward and upward subsets of the direct sum of lattice normed spaces X_{i} is discussed. This will be done by homomorphism functions. Finally, we introduce the best approximation of these sets.