Let $1 \leq p,q \leq \infty$ and $A= (a_{ij})$ be a scalar matrix. Suppose for every $x= x_j$, the series $\sum_{1}^{\infty}a_{ij}x_j$ is convergent for every $i$ and that $y=(y_i) \in {l}^q$ where $y_i = \sum_{1}^{\infty}a_{ij}x_j = (Ax)_{j}$. I need to show that the map $A : l^p \rightarrow l^q $ is a bounded linear map.
The hint given to me was to use Closed graph theorem but I don't know where to start?