# Can I use Banach-Steinhaus?

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 $$X$$.

## 1 Answer

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 $$X$$.)