On the convergence of martingales and geometric properties of Banach spaces

In this paper we give a condition which implies the almost sure convergence of a class of martingales with values in a uniformly 2-smooth Banach space. This condition is also necessary if martingales take values in a uniformly 2-convex Banach space. These results allow us to characterize the Hilbert space in terms of the almost sure convergence of martingales.