  We provide a bound on a natural distance between finitely and infinitely
supported elements of the unit sphere of $\ell^2(\mathbb{N}^*)$, the space of
real valued sequences with finite $\ell^2$ norm. We use this bound to estimate
the 2-Wasserstein distance between random variables which can be represented as
linear combinations of independent random variables. Our results are expressed
in terms of a discrepancy measure which is related to Nourdin and Peccati's
Malliavin-Stein method. The main area of application of our results is towards
the computation of quantitative rates of convergence towards elements of the
second Wiener chaos. After particularizing our bounds to this setting and
comparing them with the available literature on the subject (particularly the
Malliavin-Stein method for Variance-gamma random variables), we illustrate
their versatility by tackling three examples: chi-squared approximation for
second order $U$-statistics, asymptotics for sequences of quadratic forms and
the behavior of the generalized Rosenblatt process at extreme critical
exponent.
