<p>This paper establishes some mean convergence theorems for the maximum of normed weighted double sums of <i>M</i>-dependent and <i>M</i>-pairwise dependent Banach space valued random elements under compact uniform integrability conditions. The main results are new even when the underlying random elements are independent and the Banach space is the real line. Some results from the literature are extended from the normed single sums derived from sequences or triangular arrays of random elements to the maximum of normed double sums derived from double arrays of random elements. Furthermore, some of these results are improved by replacing the independence assumption with <i>M</i>-pairwise independence. Various examples are provided to illustrate the sharpness of the results.</p>