Why neg. cone is not bounded

Question

In a normed space X, if $S^*=\{g\in X^* : g(x)\leq 0\}$ then why $S^*$ is not bounded? Do we need to use a theorem to show that its not bounded? Any hint will be very appreciated.

Answer 1

I assume that $x$ is fixed and does not equal $0$ in your question.

Then, given $x \in X \setminus \{0\}$, there is $g \in X^*$ with $g(x) \ne 0$. W.l.o.g., we assume $g(x) < 0$. Then, it is easy to see that $g \ne 0$ and $n \, g \in S^*$ for all $n \in \mathbb{N}$. Hence, $S^*$ is not bounded.