For a double array of random elements ${X_{mn};m{geq}1,n{geq}1}$ in a $p$-uniformly smooth Banach space, ${b_{mn};m{geq}1,n{geq}1}$ is an array of positive numbers, convergence of double random series ${sum}^{infty}_{m=1}{sum}^{infty}_{n=1}X_{mn}$, ${sum}^{infty}_{m=1}{sum}^{infty}_{n=1}b^{-1}_{mn}X_{mn}$ and strong law of large numbers $$b^{-1}_{mn}sum^m_{i=1}sum^n_{j=1}X_{ij}{ ightarrow}0$$ as $$m{wedge}n{ ightarrow}{infty}$$ are established.