Absolute series convergence of linear functional values implies a kind of sequence value series convergence in Banach spaces. Ran across this question, and am not sure how to tackle it.
Let $X$ be a Banach space, and let $\{x_n\}$ be a sequence in $X$ such that $\sum_{n=1}^{\infty} |\phi(x_n)|$ converges for all $\phi \in X^*$, the dual of $X$. Show that $\sum_{n=1}^{\infty} a_n x_n$ converges in $X$ for all sequences $\{a_n\} \in c_0$, the space of all sequences converging to $0$. 

The first step in such kinds of question is often to transform the given qualitative information to a quantitative information (you will see what I mean by this). One possibility to do so is to use that closed graph theorem. Let
$$
\Phi : X^\ast \to \ell^1 ,  \phi \mapsto ( \phi (x_n))_n.
$$
By assumption, this map is well-defined. It is easy to see that it has closed graph, so that it is bounded. Hence, we have shown
$$
\sum |\phi(x_n)| \leq C
$$
for all $\|\phi\|\leq 1$.
Now, let $(a_n)_n\in c_0$ be arbitrary. We want to show that the sequence $y_n = \sum_{i=1}^n a_i x_i$ is Cauchy. Let $\epsilon>0$ and choose $N_0$ with $|a_n|\leq\epsilon$ for $n \geq N_0$. Then for $n>m>N_0$,
$$
|\phi(y_n -y_m)| =|\sum_{i=m+1}^n a_i \phi(x_i)| \leq \epsilon \sum_{i=m+1}^n |\phi (x_i)| \leq C\epsilon
$$
as soon as $\|\phi\|\leq 1$. As a consequence of the Hahn Banach theorem, this yields $\|y_n -y_m\|\leq C\epsilon$ as desired.
