BEST APPROXIMATION IN QUASI TENSORPRODUCT SPACE AND DIRECT SUM OF LATTICENORMED 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 xnvestigate 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.