Exercise 7, § 1

\* Let A be a ring with a discrete valuation, v the normed valuation of the division ring of fractions K of A; take the absolute value $a^v$ on K, where $0 < a < 1$. Let E be a normed vector space over K, for which the norm satisfies the ultrametric inequality

a) Denote by M the set of those $x \in E$ for which $\|x\| \leq 1$, and by $\pi$ a uniformizer of A; M is an A-module, and $M/\pi M$ a vector space on the residual division ring $k = A/\pi A$ of A. Let $(e_\lambda)_{\lambda \in L}$ be a family of elements of M such that the images of $e_\lambda$ in $M/\pi M$ form a basis of this vector k-space. Show that $(e_\lambda)$ is an independent family in E and that the vector subspace F of E generated by $(e_\lambda)$ is dense in E. b) If we put $\|x\|_1 = \sup_\lambda |\xi_\lambda|$, for every $x = \sum_\lambda \xi_\lambda e_\lambda$ in F, show that on F the norms $\|x\|$ and $\|x\|_1$ are equivalent. c) Let K be complete. Deduce from a) and b) that, if L is finite, the completion $\hat{E}$ of E is isomorphic to $K^L$; if L is infinite $\hat{E}$ is isomorphic to the subspace $\mathscr{C}_K^0(L)$ of $\mathscr{B}_K(L)$ formed of the families $(\xi_\lambda)$ such that $\lim \xi_\lambda = 0$ for the filter of complements of finite subsets of L. d) We suppose K and E complete; let G be a second normed complete space over K whose norm satisfies the ultrametric inequality. Show that on replacing (if necessary) the norm of $\mathscr{L}(E; G)$ (GT, X, § 3.2) by an equivalent one, then $\mathscr{L}(E; G)$ is isometric to the vector space of families $(y_\lambda)_{\lambda \in L}$ of elements of G such that $\sup_{\lambda \in L} \|y_\lambda\| < + \infty$, carrying the norm $\sup_{\lambda \in L} \|y_\lambda\|$ (which is also an ultrametric norm).