Exercise 6, § 3

Let E be a complete metrisable vector space over a non-discrete valued division ring K. a) Let $p$ be a lower semi-continuous mapping of E in the interval $[0, + \infty)$ of $\overline{\mathbf{R}}$ such that $p(\lambda x) = |\lambda|.p(x)$ for $\lambda \neq 0$ in K and $x \in E$, such that $p(0) = 0$ and $p(x + y) \leq p(x) + p(y)$ for any $x, y$ in E. Show that if $p$ is finite in E, then $p$ is continuous (consider the closed set B of $x \in E$ such that $p(x) \leq 1$, and use Baire’s theorem). b) Let $(p_n)$ be a sequence of mappings of E in $(0, +\infty)$ satisfying the conditions of a). Show that if none of the $p_n$ are finite in E then there exists a point $x \in E$ such that $p_n(x) = +\infty$ for all $n$ (same method as above).