let $X$ be a (complex) Banach space, and $\{x\_n\}$ is a sequence in $X$. Suppose that for any $f\in X'$, $$\sum\_{n=1}^\infty |f(x\_n)|<\infty.$$ Show that there exists a constant $\mu>0$ such that $$\sum\_{n=1}^\infty |f(x\_n)|\leq \mu ||f||.$$


Here, $X'$ is the Banach space consisting of all linear bounded functionals on $X$.


clearly, 
$$p(f)=\sum\_{n=1}^\infty |f(x\_n)|$$ is a sub-additive functional on $X'$. Can you use the equivalent norm theorem on Banch space?


I have not get a solution.



On the dual of Banach space
You can prove the desired as follows. Consider the map from $X'$ to $\ell\_1$ defined by $f\mapsto \{f(x\_n)\}\_{n=1}^\infty$. Show that it satisfies the assumptions of the Closed Graph Theorem, conclude that it is a continuous operator. The desired inequality follows.


