Let $X$ be an infinte-dimensional normed space. Let $\ell_1,\ldots, \ell_n$ be continuous linear functionals on $X$ and consider the set $U = \{x\in X : |\ell_j(x)| < 1,\;\; 1\leq j \leq n\}.$ Show that the set $U$ is unbounded.
If we put the functionals together to a linear map $L:X\rightarrow \mathbb{C}$, $L(x) = (\ell_1(x), \ell_2(x),\ldots,\ell_n(x))$. The kernel of this map is a subspace, right? How would this kernel look? would it be anything else then $x= \textbf{0}$? I seem to be stuck here.