Suppose $(w_{\lambda})_{\lambda\in I}$ and $(\nu_{\lambda})_{\lambda\in I}$ are orthonormal bases of a Hilbert module $E$ over the C*_algebra $\mathcal{A}$, such that $w_{\lambda}=u(\nu_{\lambda})$ where $u$ is a unitary operator on the dense submodule of $E$ . We define $U:E\to E$ $~$by $$U(x)=\sum_{\lambda} w_{\lambda}\langle \nu_{\lambda},x\rangle,$$
Why is $U$ well defined?
If you are taking about the convergence of the series, the coefficients are in $\ell_2$ because of Parseval's Identity. And the norms of the tails of your series are precisely the tails of Parseval's.