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.