Solution
= Solution
Define
$$
T:V^*\to\ell^1,
\qquad
Tf=(f(x_n))_{n\geq1}.
$$
The hypothesis makes $T$ everywhere defined. The dual $V^*$ is Banach even when $V$ is incomplete. If $f_j\to f$ in $V^*$ and $Tf_j\to a$ in $\ell^1$, then coordinatewise convergence gives $a_n=f(x_n)$, so the graph of $T$ is closed. The <closed graph theorem> therefore gives
$$
\|Tf\|_1\leq C\|f\|_{V^*}.
$$
Taking the supremum over the unit ball proves the required finite bound.