On the theory of singular expansion in a tensor product of Hilbert spaces

Approximation of an element in a tensor product of Hilbert spaces by a sum of products of elements in each of these spaces is considered. The existence of best approximation and the convergence of bilinear series are proved. An explicit representation for the best approximation is obtained. Analogous results in the Sobolev spaces $W_2^{p,q}(\Pi)$ are given as corollaries.