Well, it is a bit too strong for you, but should be noted that if $a_{nm}$ converges uniformly in $m$, and your $L_2$ exists, then all the three limits exist and are equal. This is a classical result having generalizations to net-convergence, and its first version is attributed to Cauchy.
Well, it is a bit too strong for you, but should be noted that if $a_{nm}$ converges uniformly in $m$, then all the three limits exist and are equal. This is a classical result having generalizations to net-convergence, and its first version is attributed to Cauchy.