characterization of an infinite matrix mapping and continuity Show that an infinite matrix mapping $A=[a_{ij}]$ $:l^{\infty}\to l^{\infty}$ is continuous iff $sup_{i\in \mathbb N}$ $ \sum_{k=1}^{\infty}{|a_{ij}|}=||A||<\infty$.  Give a characterization of the matrix which map $c_0$ continuously into $c_0$
Where $c_0$ denotes the set of all sequences that converges to zero, as a subspace of $l^{\infty}$ endowed with the sup-norm.
 A: Fix $i\in\mathbb{N}$ and consider $x^{(i)}\in \ell_\infty$ such that $x_j^{(i)}=\operatorname{sign}(a_{ij})$. Then $\Vert x^{(i)}\Vert=1$ and 
$$
\sum\limits_{j=1}^\infty|a_{ij}|
=\left|\sum\limits_{j=1}^\infty a_{ij} x_j^{(i)}\right|
\leq\Vert A(x^{(i)}) \Vert\leq\Vert A\Vert\Vert x^{(i)}\Vert=\Vert A\Vert
$$
Since $i$ is arbitrary we get $\sup_{i\in\mathbb{N}} \sum_{j=1}^\infty|a_{ij}|\leq\Vert A\Vert$. On the other hand for arbitrary $x\in\ell_\infty$ we have
$$
\Vert A(x)\Vert
=\sup_{i\in\mathbb{N}}\left|\sum_{j=1}^\infty a_{ij}x_j\right|
\leq\sup_{i\in\mathbb{N}}\sum_{j=1}^\infty |a_{ij}|\sup_{j\in\mathbb{N}}|x_j|
=\left(\sup_{i\in\mathbb{N}}\sum_{j=1}^\infty |a_{ij}|\right)\Vert x\Vert
$$
so $\Vert A\Vert\leq \sup_{i\in\mathbb{N}}\sum_{j=1}^\infty |a_{ij}|$. From inequalities obtained we get the desired equality. 
As for the second question. Necessary and sufficient condition is $A(e_k)\in c_0$ for all $k\in\mathbb{N}$. Indeed, in this case $\operatorname{span}\{A(e_k):k\in\mathbb{N}\}\subset c_0$, so
$$
A(c_0)
=A(\overline{\operatorname{span}\{e_k:k\in\mathbb{N}\}})
\subset \overline{\operatorname{span}\{A(e_k):k\in\mathbb{N}\}}
=\overline{c_0}
=c_0
$$
Conversely, if $A(c_0)\subset c_0$, then obviously $A(e_i)\in c_0$ for all $i\in\mathbb{N}$. It is remains to interpret condition $A(e_k)\in c_0$ in terms of $[a_{ij}]$:
$$
0
=\lim\limits_{i\to\infty} A(e_k)_i
=\lim\limits_{i\to\infty} \sum\limits_{j=1}^\infty a_{ij}\delta_{kj}
=\lim\limits_{i\to\infty} a_{ik}
$$
