I was wondering if there is a characterization for those double sequences $\{b_{n,j}\in \mathbb{C}~\vert~n\in \mathbb{N},~j\in \{0,...,n\}\}$ such that for all bounded sequences $s_j\in \mathbb{C}$, we have that $$\sigma_n =\sum_{j=0}^n b_{n,j}s_j$$ converges?
Note I do not ask that for if $s_j\rightarrow s$, then $\sigma_j\rightarrow s$, like in usual summability methods.
Some thoughts: In functional analysis language, I am basically embedding $c_{00}$ in $ba = (\ell^\infty)^*$ and asking how weak*-convergence of sequences behaves. Maybe this point of view could come in useful.