On Polar Sets with respect to Continuous Seminorms In the following, $X$ is a Hausdorff locally convex topological vector space and $X'$ is the topological dual of $X$. If $p$ is a continuous seminorm on $X$ then we shall designate by $U_p$ the "$p$-unit ball", i.e,
$$U_p=\{x\in X: p(x)\le 1\}.$$
The polar set of $U_p$ is given by
$$U_p^o=\{f\in X':|f(x)|\le 1\quad \forall x\in U_p\}.$$
How do we prove that for each $x\in X$, we have
$$p(x)=\sup\{|f(x)|:f\in U_p^o\}.$$
I need some help...Thanks in advance.

Assume that $p(x)=0$. Then for all $\lambda>0$, $\lambda x\in U_p$ hence $|f(\lambda x)|\leqslant 1$ and $f(x)=0$ whenver $f\in U_p^0$. 
If $p(x)\neq 0$, then considering $\frac 1{p(x)}x$, we get $\geqslant$ direction. For the other one, take $f(a\cdot x):=a\cdot p(x)$ for $a\in\Bbb R$; then $|f(v)|\leqslant p(v)$ for any $v\in\Bbb R\cdot x$. We extend $f$ by Hahn-Banach theorem to the whose space: then $|f(w)|\leqslant p(w)$ for any $w\in X$, giving what we want.
