 Solution

DefineThe hypothesis makes $T$ everywhere defined. The dual $V^{\ast }$ is Banach even when $V$ is incomplete. If $f_j\to f$ in $V^{\ast }$ and $T{f}_j\to a$ in ${\ell }^1$, then coordinatewise convergence gives $a_n=f(x_n)$, so the graph of $T$ is closed. The closed graph theorem therefore givesTaking the supremum over the unit ball proves the required finite bound.