On the rate of convergence in the central limit theorem in some Banach spaces

Let $B$ be a real separable Banach space and $\xi_i$, $i=1,2,\dots,n$, be independent random variables with values in $B$ and $\mathbf E\xi_i=0$, $\mathbf E\|\xi_i\|^3=0$. Under some conditions on the space $B$, we estimate closeness between the distrubutions of the normalized sums $\displaystyle B_n^{-1}\sum_{i=1}^n\xi_i$ and Gaussian distributions on $B$. In Theorem 1, a general estimate is given. In Theorem 2, when the summands are identically distributed, a better estimate is obtained. It is worth mentioning that, even in the case of a real separable Hilbert space, this estimate is new.