Let $w$ be the space of sucessions with coefficients in $\mathbb{K}$, with $\mathbb{K}= \mathbb{R}$ or $\mathbb{C}$. For $x \in w$ ,let $p_n(x)=\max_{1 \leq i \leq n} |x_i|$ be a seminorm and consider $w$ as a locally convex space with the family of seminorms $(p_n)_n$.
Let $A=(a_{ij})_{i,j \in \mathbb{N}}$ be an infinite matrix and let $T: w \rightarrow w$ be the map $Tx=Ax=(\sum_{j=1}^\infty a_{ij}x_j)_i$. When is $T$ continuous?
I have showed in another exercise that for any $i\in \mathbb{N}$ we need the existence of a natural $j_i$ so that $a_{ij}=0$ for all $j>j_i$, and using that I showed that given $n \in \mathbb{N}$, there exist a constant $\lambda=max_{1\leq i\leq n} \sum_{j=1}^{j_i} |a_{ij}|$ and a natural $M=\max_{1\leq i \leq n} j_i$ so that $p_n(Tx)\leq \lambda p_M(x)$, but I didn't impose nothing else on the matrix. I don't know if this is correct or if I should have to impose more restrictions on $A$.
Can anyone help me?