Need help with this problem on the applications of the Hahn-Banach Theorem for normed spaces. If $p$ is a defined on a vector space $X$ and satisfies the properties
$p(x+y) \le p(x) + p(y)$ and $p(\alpha x) = |\alpha|p(x)$,
how do I show that for any $x_0 \in X$ there exists some linear functional $\bar f$ such that $\bar f(x_0) = p(x_0)$ and $|\bar f(x)| \le p(x)$ for all $x \in X$?

Start with the one dimensional subspace $W$ spanned by $x_0$ and define $f:W\to k$ by
$$
f(\alpha x_0) = \alpha p(x_0)
$$
This is a well-defined linear functional that satisfies $|f(x)| \leq p(x)$ for all $x\in W$. Now simply apply Hahn-Banach to get $\overline{f}$ defined on the whole space.
