Exercise 8, § 3

a) Let K be a non-discrete valued division ring of which the absolute value is an ultrametric, non linearly compact (cf. CA, III, § 2, exerc. 15); then there exists a well ordered set I of numbers $> 0$ and a family $(B(\rho))_{\rho \in I}$ of closed balls in K such that the relation $\rho < \rho'$ implies $B(\rho) \subset B(\rho')$, that $B(\rho)$ has radius $\rho$ and that the intersection of the $B(\rho)$ is empty (CA, VI, § 5, exerc. 5). For every $x \in K$, there exists $\rho \in I$ such that $x \notin B(\rho)$; show that the number $\phi(x) =$ $|x - y|$ for $a\ y \in B(\rho)$ depends neither on $y \in B(\rho)$, nor on $\rho \in I$ such that $x \notin B(\rho)$. If $\rho \in I$ is such that $x \in B(\rho)$ then $\phi(x) \leq \rho$. This being so, for $(x_1, x_2) \in K^2$, put $\|(x_1, x_2)\| = |x_1|$ if $x_2 = 0$, and $\|(x_1, x_2)\| = |x_2| \ \phi(x_2^{-1} x_1)$ if $x_2 \neq 0$. Show that $\|(x_1, x_2)\|$ is an ultranorm on $K^2$ (I, p. 26, exerc. 12) and show that there does not exist any projection of norm 1 of $K^2$ on $K \times \{0\}$.

b) Let $K$ be a complete non-discrete valued division ring of which the absolute value is an ultrametric and which is linearly compact. Let $E$ be a vector space of dimension 2 on $K$ with an ultranorm and let $D$ be a line in $E$; show that for all points $x \in E$, there exists $y \in D$ such that $d(x, D) = d(x, y) = \|x - y\|$ (note that the intersection of $D$ and of a ball of centre $x$ is a ball in $D$).

c) Deduce from a) and b) that for a complete non-discrete valued division ring $K$, of which the absolute value is an ultrametric, the following properties are equivalent :

$\alpha)$ $K$ is linearly compact.

$\beta)$ For every ultranormed vector space $E$ on $K$, for every vector subspace $F$ of $E$ and every continuous linear form $f$ on $F$ there exists a continuous linear form $g$ on $E$ that extends $f$ and is such that $\|g\| = \|f\|$. (Reduce to the case where $E$ is of dimension 2 and use b).) \*