Exercise 43, § 2

Let K be a complete valued division ring of which the absolute value is an ultrametric. In a left vector space E on K, we say that a set A is ultraconvex if the relations $x \in A, y \in A, |\lambda| \leq 1, |\mu| \leq 1$ imply $\lambda x + \mu y \in A$. a) Generalize the prop. 1, 2, 5, 6, 7 of II, p. 8 and p. 9. Show that the smallest ultraconvex set containing a given set M is the set of linear combinations $\sum \lambda_i x_i$, where $x_i \in M$ and $|\lambda_i| \leq 1$ for all i. b) Suppose that E is a topological vector space over K. Show that the closure of an ultraconvex set is ultraconvex, and that an ultraconvex set with a non-empty interior is open. c) Let A be an absorbent and ultraconvex set in E. Show that if, for all $x \in E$, we put $p(x) = \inf_{x \in \rho A} |\rho|$, then p is an ultra-semi-norm on E (II, p. 2). Generalize prop. 23 of II, p. 20, to the case where the absolute value of K is obtained from a discrete valuation.