How to take out the summation $\sum\limits_{l=1}^{\infty}$ out of the $2$-norm?
Consider the $2$-norm of the double summation $\left\|\sum\limits_{l=1}^\infty\sum\limits_{k=1}^\infty a_{kl} x_{kl}\right\|_2$, where $a_{kl}$ are the scalars.
I want to decompose with respect to the $j$ by bringing out the summation $\sum_{l=1}^\infty$ out of the norm symbol.
We know by definition of $2$-norm, $\|x\|_2^2=\sum\limits_{k=1}^\infty|x_k|^2$, for $x=(x_1,~x_2,\ldots)$.
How to take out the summation $\sum\limits_{l=1}^\infty$ out of the norm ??