# Double Sequence Defines a Bounded Linear Map on l2

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