= Solution
For finite subsets $F\subset\mathbb Z$, orthonormality gives
$$
\left\|\sum_{n\in F}\alpha_ne_n\right\|^2=\sum_{n\in F}|\alpha_n|^2.
$$
For $S_N=\sum_{|n|\leq N}\alpha_ne_n$, 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 \b[the bilateral orthogonal series converges in $X$ independently of its ordering]. This is the <Riesz-Fischer theorem for an orthonormal system>.
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