As an application, we derive a result (Theorem 111.13) of the form of the first proposition above which is valid for all normed linear spaces and in which R restricts only the variances of the random variables. This theorem is best possible in the sense indicated there. Theorem 111.5 depends on the fact that in certain families of independent sequences of random variables such that each sequence averages to 0 almost surely the speed of convergence is uniform. ?1 is devoted to introducing the necessary concepts for sequences of functions and deriving uniform speed of convergence