= Solution
Apply <continuity> of the <inner product> to the convergent expansion of $x$:
$$
\boxed{\langle x,y\rangle=\sum_{n\in\mathbb Z}\langle x,e_n\rangle\langle e_n,y\rangle.}
$$
The series is absolutely convergent because the <Cauchy-Schwarz inequality> for scalar sequences and <Bessel's inequality> give
$$
\sum_n|\langle x,e_n\rangle\langle e_n,y\rangle|\leq\left(\sum_n|\langle x,e_n\rangle|^2\right)^{1/2}\left(\sum_n|\langle y,e_n\rangle|^2\right)^{1/2}\leq\|x\|\|y\|.
$$
This is the polarized <Parseval identity>.
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