Suppose that $(X, \|.\|)$ is an infinite dimensional Banach space. I would like to ask whether we could construct a sequence $\{x_n^*\}_{n\in \mathbb{N}}\subset X^*$ (dual space of $X$) such that:
$\|x_n^*\|_{X^*}=1$;
$\{x_n\}$ is weakly convergent to $0$.
Example. Let $H=\ell_2$ be the real Hilbert space. Then $H^*=\ell_2$ and we can choose $\{x^n\}_{n\in\mathbb{N}}\subset H^*$ as $ x^1=(1,0, \ldots, 0) $ $ x^2=(0,1, \ldots, 0) $ $ \vdots $ $ x^n=(0,\ldots,0,1,0,\ldots, 0) $ $ \vdots $ Then $x^n\overset{w^*}{\rightarrow} 0$ and $\|x^n\|_{H^*}=1$.