A bounded sequence in a Banach space

Question

Let $X$ be a Banach space and $\langle x_n\rangle $ be a sequence in $X$.

If ( $f(x_n)$ ) is a bounded sequence for any bounded linear functional $f$ on $X$, then ( $x_n$ ) is a bounded sequence in $X$.

I have to prove this fact. I first thought it would be simple, but it turns out to be trickier...

Could anyone help me with this?

Answer 1

As PhoemueX said: the statement follows from the Uniform Boundedness Principle applied to linear maps $\phi_n:X^*\to \mathbb{K}$ that are defined by $\phi_n(f)=f(x_n)$. ($\mathbb{K}$ stands for $\mathbb{R}$ or $\mathbb{C}$.) The principle applies here because $X^*$ is complete even when $X$ isn't.