The purpose of this note is to present a certain necessary and sufficient condition for strong convergence in Lp over a product space. This result is formulated as Theorem 1 below. Theorem 1 includes as a special case a well-known compactness condition of M. Riesz, which we state in slightly different form as Theorem 2. In the final part of the paper we show that Theorem 1 can be used to give very simple proofs of the so-called Rellich lemma of functional analysis.