Given $p\in (1,\infty)$. Take a bounded sequence $(f_n)$ in $\ell^p$ and define a linear map $T\colon \ell^p_{00}\to \ell^p$ ($\ell^p_{00}$ is the space of finitelty supported sequences in $\ell^p$) by
$Te_n = f_n$
($e_n$ is the usual (Hamel) basis for $\ell^p_{00}$).
Is $T$ a bounded linear map? If $p=1$, the answer is yes... If not, what do have to assume on $(f_n)$ to get boundeness of $T$?