Lauricella's theorem

In the theory of orthogonal functions, Lauricella's theorem provides a condition for checking the closure of a set of orthogonal functions, namely:

"Theorem." A necessary and sufficient condition that a normal orthogonal set $\{u\_k\}$ be closed is that the formal series for each function of a known closed normal orthogonal set $\{v\_k\}$ in terms of $\{u\_k\}$ converge in the mean to that function.

The theorem was proved by Giuseppe Lauricella in 1912.

Reference

*G. Lauricella: "Sulla chiusura dei sistemi di funzioni ortogonali", Rendiconti dei Lincei, Series 5, Vol. 21 (1912), pp. 675–85.