The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L²(X × Y) by sums_k=1ⁿf_kf_k, with a prescribed numbern of products of arbitrary functionsf_k L²(X) andg_k L²(Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL²(X × Y).