A sequence $\{ \varphi_{k} \}$ of real functions, complete in the $\mathfrak{L}^{2}(T)$ space, relative to a set $T$ of the euclidean r-space, is considered. A condition is given for the completeness of a subsequence $\{ \psi_{k} \}$ of $\{ \varphi_{k} \}$ in $\mathfrak{L}^{2}(U)$ ($U$ subset of $T$).