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.