About absolute convergent series in Banach space and its bounded linear functional.

Question

Let $\{x_n\}$ be a sequence in Banach space $X$ such that $\sum_{n=1}^{\infty} |f(x_n)|< \infty$ for all bounded linear functional $f \in X'$.

Show that there exists $M\geq 0$ such that $\sum_{n=1}^{\infty} |f(x_n)| \leq M \|f\|$, for all $f\in X'$.

What I have tried:

Since $|f(x_n)| \leq \|f\| \| x_n\|$, if we can obtain $\sum_{n=1}^{\infty} \|x_n\| \leq M$, the proof can be got. But I don't know in what condition a sequence in Banach space will be absolute convergent.

And from the condition we can know that $f(\sum_{n=1}^{\infty} x_n) < \infty$.

Answer 1

Define $T_n:X'\to l^1(\mathbb C)$ by $$T_n(f)=(f(x_1),f(x_2),...,f(x_n),0,0,0...),\;f\in X'.$$ For each $f$ the sequence $\{\|T_n(f)\|_{l^1}\}_n$ is bounded. Now apply Uniform boundedness principle.