Absolute convergence of a double sum.

Question

If $A\in\mathbb{C}^{\infty\times \infty}$ such that

$$ \sum_{i=0}^\infty \left|\sum_{j=0}^\infty a_{ij}\right|<\infty,\qquad \sum_{j=0}^\infty \left|\sum_{i=0}^\infty a_{ij}\right|<\infty,$$

then is it true that

$$ \sum_{i=0}^\infty\sum_{j=0}^\infty \left|a_{ij}\right|<\infty?$$

Answer 1

No. Take $a_{i,j}=x_i y_j$ with two convergent but not absolutely convergent series such that $\sum\limits_{i=0}^\infty x_i = \sum\limits_{j=0}^\infty y_j =0$.