Exercise 10, § 2

Let K be a locally compact non-discrete valued division ring, and E a left vector space of finite dimension over K. Denote by $\mathfrak{N}(E)$ the set of norms on E, which is a subspace of the space $\mathscr{C}(E; \mathbf{R})$ of mappings of E, continuous (in the canonical topology), in $\mathbf{R}$. a) When we give to $\mathscr{C}(E; \mathbf{R})$ the compact convergence topology \* (for which it is a Fréchet space) *, the set $\mathfrak{N}(E)$ is closed in $\mathscr{C}(E; \mathbf{R})$, and locally compact. b) Let $p_0$ be an element of $\mathfrak{N}(E)$; show that there exists a continuous mapping $(\lambda, p) \mapsto \pi_\lambda(p)$ of $[0, 1] \times \mathfrak{N}(E)$ in $\mathfrak{N}(E)$ such that $\pi_0(p) = p$ and $\pi_1(p) = p_0$ for every $p \in \mathfrak{N}(E)$.