For a double array of random elements {$V_{mn};m{geq}1,;n{geq}1$} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which $k_{mn}^{-frac{1}{r}}sum{{u_m}atop{i=1}}sum{{u_n}atop{i=1}}(V_{ij}-E(V_{ij}|F_{ij})){ ightarrow}0$ in $L_r$ (0 < r < 2). The weak law results provide conditions for $k_{mn}^{-frac{1}{r}}sum{{T_m}atop{i=1}}sum{{ au}_natop{j=1}}(V_{ij}-E(V_{ij}|F_{ij})){ ightarrow}0$ in probability where {$T_m;m;{geq}1$} and {${ au}_n;n;{geq}1$} are sequences of positive integer-valued random variables, {$k_{mn};m{{geq}}1,;n{geq}1$} is an array of positive integers. The sharpness of the results is illustrated by examples.