Suppose $a_{m,n}$ is a double sequence in $\mathbb{C}$ such that both $\sup_n \sum_m |a_{m,n}|$ and $\sup_m \sum_n |a_{m,n}|$ are finite. Prove that the map $T$ defined by $$\forall s \in \ell^2, (T(s))_n = \sum_m a_{m,n} b_m$$ is a bounded linear map on $\ell^2$.

My thoughts: linearity is obvious, and I need to show the boundedness. I can show that $\left\Vert T(s)\right\Vert \leq \left\Vert s\right\Vert \sum_n (\sup_m|a_{m,n}|)^2$, but I cannot show that $\sum_n (\sup_m|a_{m,n}|)^2$ is finite......

Any help or hint is appreciated


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