Exercise 12, § 2

Let K be a locally compact valued division ring whose absolute value is non-discrete and ultrametric. We call a norm on the left vector space E over K an ultranorm if it satisfies the ultrametric inequality (II, p. 2). a) Let E be a finite dimensional left vector space over K, let $\alpha$ be an ultranorm on E and H a hyperplane in E given by the equation $\langle x, a^* \rangle = 0$. Show that there exists a point $x_0 \in E$ at which the function $x \mapsto |\langle x, a^* \rangle|/\alpha(x)$ attains its upper bound in $E \setminus \{0\}$; show that then

Deduce that there exists a basis $(a_i)$ of E and a family $(r_i)$ of real numbers $> 0$ such that, for all $x = \sum_i \xi_i a_i$ we have $\alpha(x) = \sup_i (r_i |\xi_i|)$. We say that $\alpha$ is in the standard form relative to the basis $(a_i)$. b) Let $\alpha^*$ be the norm on $E^*$ the dual of E canonically associated with $\alpha$ by it is an ultranorm. Show that for all $x_0 \neq 0$ in E, there exists $x_0^* \in E^*$ such that $\alpha(x_0) = |\langle x_0, x_0^* \rangle|/\alpha^*(x_0^*)$. c) Let $\alpha, \beta$ be any two ultranorms on E. Show that there exists a basis of E such that relative to this basis $\alpha$ and $\beta$ are both of the standard form (consider a point $x_0 \in E \setminus \{0\}$ at which $\alpha/\beta$ attains its maximum; then use b) and proceed by induction on dim E). d) Let $\mathfrak{N}_0(E)$, the set of ultranorms on E, be considered as a subspace of $\mathfrak{N}(E)$ (exerc. 10). Show that $\mathfrak{N}_0(E)$ is closed in $\mathfrak{N}(E)$. Let $\alpha_0$ be an element of $\mathfrak{N}_0(E)$; for each $\alpha \in \mathfrak{N}_0(E)$ and for $0 \leq t \leq 1$, let $P_\alpha(t)$ be the set of $\beta \in \mathfrak{N}_0(E)$ such that $\beta(x) \leq \alpha_0(x)^{1-t} \alpha(x)^t$ for all $x \in E$. Show that $P_\alpha(t)$ is not empty and that $\pi_t^\alpha = \sup P_\alpha(t)$ is an ultranorm. Further, the mapping (t, α) ↦ π_t^α of [0, 1] × Ω_0(E) in Ω_0(E) is continuous and such that π_0^α = α_0 and π_1^α = α (use c)).

e) Let A be the ring of the absolute value of K, m its maximal ideal such that k = A/m is a finite field with q elements (CA, VI, § 5, No. 1, prop. 2). For every ultranorm α on E, the image Xα of the set of values of log α(x) for x ∈ E \ {0} under the canonical mapping in the quotient group R/(Z.log q) is a finite set having at most n = dim E elements (use a)); the number of these elements is denoted by r(α) and called the rank of α. Show that r is a lower semi-continuous mapping of Ω_0(E) in N and that the set Ω'_0(E) of the α for which r(α) = n is open and everywhere dense in Ω_0(E) (use a) and c)).

f) Suppose that r(α) = n; let (a_i) be a basis of E relative to which α is of the standard form; show that there exists a neighbourhood V of α in Ω'_0(E) such that every β ∈ V has the standard form relative to (a_i) (use b)); deduce that there is a neighbourhood W ⊂ V of α homeomorphic to an open set in R^n.

g) For every basis (a_i) of E, show that the set of ultranorms α that have a standard form relative to (a_i) is closed in Ω_0(E). Deduce that if α ∈ Ω'_0(E) has a standard form relative to (a_i) the same is true of all elements of the connected component containing α in Ω'_0(E). \*