Can I use Banach-Steinhaus?

Question

Let $X$ be a Banach space and $(x_n)_n$ a subset of $X$ such that$$\sum_{j\geq1}|\langle\phi,x_j\rangle|<∞.\quad\forall\phi\in X^*$$Show that$$\sup_{\|\phi\|=1}\sum_{j\geq1}|\langle\phi,x_j\rangle|<∞.$$

Can I use Banach-Steinhaus to show it quickly? I am not pretty sure cause I do not have a magre subset of $X$.

Answer 1

Hint: Yes, apply Banach-Steinhaus for $\{\phi\mapsto\langle\phi,x_n\rangle:n\in\Bbb N\}\ \subseteq\ \mathcal B(X^*,\Bbb C)$.
(It requires $X^*$ to be Banach, which is true for any normed space $X$.)