(where
denotes the L2-norm with respect to a weighting
function )
converges to zero as becomes infinite. Symbolically, a set of functions is complete
if

Arfken, G. "Completeness of Eigenfunctions." §9.4 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 523-538,
1985.Kaplan, W. "Fourier Series of Orthogonal Functions: Completeness"
and "Sufficient Conditions for Completeness." §7.11 and 7.12 in Advanced
Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 501-505, 1992.