  A general device is proposed, which provides for extension of exponential
inequalities for sums of independent real-valued random variables to those for
martingales in the 2-smooth Banach spaces. This is used to obtain optimum
bounds of the Rosenthal-Burkholder and Chung types on moments of the
martingales in the 2-smooth Banach spaces. In turn, it leads to best-order
bounds on moments of the sums of independent random vectors in any separable
Banach spaces. Although the emphasis is put on the infinite-dimensional
martingales, most of the results seem to be new even for the one-dimensional
ones. Moreover, the bounds on the Rosenthal-Burkholder type of moments seem to
be to certain extent new even for the sums of independent real-valued random
variables. Analogous inequalities for (one-dimensional) supermartingales are
given.
