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$U=R^1$.