best approximation in quasi tensor product space and direct sum of lattice normed spaces

we study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesx_{i}. we introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by x boxtimesy. we investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor 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 withdownward and upward subsets of the direct sum of lattice normedspaces x_{i} is discussed. this will be done by homomorphismfunctions. finally, we introduce the best approximation of thesesets.