Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces

The problem under consideration is to estimate the distance, with respect to a chosen metric $\mu$, between two linear combinations $\displaystyle X=\sum_jc_jX_j$ and $\displaystyle Y=\sum_jc_jY_j$ of independent random variables with values in a Banach space $U$.
General results of this paper enable, in particular, to effectively estimate the accuracy of approximation of the distributions of normalized sums of independent random $U$-valued variables by a normal law.
When choosing $\mu$ in an appropriate way, one obtains estimates quite analogous to those known in the simplest case $U=R^1$.