Upper bounds for the concentration function in a Hilbert space

New bounds (analogous to the bounds obtained by Kolmogorov, Rogozin and Esseen) are derived for the concentration function of the sums of independent random variables with values in a Hilbert space. In particular, the absolute constants used in the estimates don't depend on the dimension in the finite-dimensional space. Further, some limit theorems for the concentration function and some estimates for the concentration functions of infinitely divisible distributions are given.