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.

References

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

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