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

''Theorem.'' A necessary and sufficient condition that a normal orthogonal set <math>\{u_k\}</math> be closed is that the formal series for each function of a known closed normal orthogonal set <math>\{v_k\}</math> in terms of <math>\{u_k\}</math> [[Convergence_of_random_variables#Convergence_in_mean|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.

[[Category:Functional analysis]]
[[Category:Theorems in functional analysis]]

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