 Solution

For finite subsets $F\subset \mathbb{Z}$, orthonormality givesFor $S_N={\sum }_{|n|\le N}{\alpha }_n{e}_n$, the squared norm of $S_N-S_M$ is the corresponding finite tail sum. Since ${\sum }_{n\in \mathbb{Z}}|{\alpha }_n{|}^2<\infty$, these tails tend to zero. Thus $(S_N)$ is a Cauchy sequence in the complete Hilbert space and converges.

The same estimate for arbitrary finite tails proves that the sum is unconditional, so the bilateral orthogonal series converges in $X$ independently of its ordering. This is the Riesz-Fischer theorem for an orthonormal system.