BEST APPROXIMATION IN QUASI TENSOR PRODUCT SPACE AND DIRECT SUM OF LATTICE NORMED SPACES

We study the thoery of best approximation in tensor product space and the direct sum of some lattice normed spaces Xi: 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?Y. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessar- ily downward or upward and we call them Im??quasi downward or Im??quasi upward. We show that these sets can be interpreted as downward or upward sets. The relation of these sets with down- ward and upward subsets of the direct sum of lattice normed spaces Xi is discussed. This will be done by homomorphism functions. Fi- nally, we introduce the best approximation of these sets.