Estimates of the accuracy of normal approximation in a Hilbert space

Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables with values in a separable Hilbert space such that $\mathbf EX_j=0$, $\mathbf E|x_j|^{3+\delta}<\infty$, $0\le\delta\le 1$. Estimates of the accuracy of normal approximation for $\mathbf P\{|n^{-1/2}(X_1+\dots+X_n)|<r\}$ are constructed. For $0\le\delta\le 1$ the order of approximation is $O(n^{-1_+\delta)/2})$, for $\delta=1$ the order is $O(n^{-1+\varepsilon})$, $\varepsilon>0$.