Why is $\sum_a u_a(y)u_a^*(x)=\delta(x-y)$ (the completeness relation) equivalent to saying that for any arbitrary $f(x)$ s.t. $\int_{-\infty}^{\infty}|f(x)| = 1$, $f(x)$ can be expanded as $\sum_ac_au_a(x)$, where $c_a$'s are constants (the expansion theorem)?
Thanks.
Symbols:
{$u_a(x)$} are orthonormal eigenfunctions
$\delta(x-y)$ is the Dirac delta function