Abstract: In this paper, we obtain the almost sure bound of normal sum of B-valued weakly dependent random variables under the assumption that Banach space B is p-smoothable (1<p\leq2). As its application, we characterize p-smoothness of Banach space through the strong law of large numbers of B-valued y-mixing random variables, and obtain some results on the strong law of large numbers for B-valued dependent random variables without assumption on rate of tending to zero of \varphi and \psi-mixing parameters \varphi_n and \psi_n but under the assumption that \inf\limits_n\geq1\varphi_n=0 or \inf\limits_n\geq1\psi_n=0 respectively.