From Steele's Cauchy-Schwarz Master Class I'm having a hard time understanding the solution problem 1.13:
Show that if {$a_{jk}$: 1 ≤ j ≤ m, 1 ≤ k ≤ n} is an array of real numbers then one has
$$m\sum_{j=1}^m\left(\sum_{k=1}^na_{jk}\right)^2 + n\sum_{k=1}^n\left(\sum_{j=1}^ma_{jk}\right)^2 \leq \left(\sum_{j=1}^m\sum_{k=1}^na_{jk}\right)^2 + mn\sum_{j=1}^m\sum_{k=1}^na_{jk}^2$$
The solution outline: "If we view {$x_{jk}$ : 1 ≤ j ≤ m, 1 ≤ k ≤ n} as a vector of length mn then Cauchy’s inequality and the one-trick splitting xjk = xjk · 1 imply the general bound
($\sum_{j=1}^m\sum_{k=1}^nx_{jk})^2$ ≤ mn $\sum_{j=1}^m\sum_{k=1}^nx_{jk}^2$
We apply this bound to $x_{jk}$ = $a_{jk}$ − $r_j$/n − $c_k$/m where $r_j$ = $\sum_{k=1}^na_{jk}$ and $c_k = \sum_{j=1}^ma_{jk}$ and if we set T = $\sum_{j=1}^m\sum_{k=1}^na_{jk}$
We are suppose to get $T^2$ for the left side of the bound I don't follow the summation for this part.