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.