On the Minimality of Projections in Expanded Matrix Spaces

Let S be a space of all functions from X×Y×Z into R or C, where X,Y,Z are finite sets. It was proved that there is a unique projection from S into its subspace consisting of all sums of functions that depend on one variable. We show here its explicit formula. A similar fact was proved for the subspace of S consisting of all the sums of functions that depend on two variables. We prove a significant generalization of this result, considering a space of all functions from X1×X2×…×Xn and its subspace consisting of all sums of functions that depend on n−1 variables, where n is any natural number. We also show that these results are true not only for Lp-spaces but also for many others.