Let $X$ be a Banach space and let $(x_n)_{n=1}^\infty$ be a sequence in $X$ such that, for some $x \in X$,
$$ \lim_{n \rightarrow \infty} \ell(x_n) = \ell(x) \text{ for all bounded linear functional } \ell : X \rightarrow \mathbb{R}. $$ I'm trying to show that $ \|x \| \leq \liminf_{n\rightarrow\infty } \|x_n\|$.
By using definition of norm of linear functional, we can get two inequality.
$\| \ell(x_n) \| \leq \| \ell \| \|x_n\|$, $\| \ell(x) \| \leq \| \ell \| \|x\|$.
We know that $\ell(x_n) \rightarrow \ell(x)$, so $\|\ell(x) \| \leq \|\ell\| \liminf_{n \rightarrow \infty} \|x_n\|$.
Do I need to show that $\|x \| \leq \frac{\|\ell(x)\|}{\|\ell\|}$ to solve this problem?
I think I do not use the conditions(Banach space or *all bounded linear functional) appropriately.