Exercise 14, § 1

Let K be a Hausdorff topological division ring, $(E_\iota)_{\iota \in I}$ an infinite family of Hausdorff topological vector spaces on K, none of which is the single point 0. We consider on $F = \prod_{\iota \in I} E_\iota$ the topology $\mathscr{T}$, compatible with the additive group structure of F, for which a fundamental system of neighbourhoods of 0 is formed by the products $\prod_{\iota \in I} V_\iota$, where, for each $\iota \in I$, the set $V_\iota$ is a neighbourhood of 0 in E (this topology is strictly finer than the product topology; cf. GT, III, § 2, exerc. 23). We denote by $\mathscr{T}_0$ the topology induced by $\mathscr{T}$ on the subspace $E = \bigoplus_{\iota \in I} E_\iota$ of F; E is closed in F for the topology $\mathscr{T}$, and if each of the $E_\iota$ is complete, then F is complete for the topology $\mathscr{T}$, therefore E is complete for the topology $\mathscr{T}_0$ (GT, III, § 3, exerc. 10). a) Show that if there exists in K a neighbourhood of 0 bounded on the right (GT, III, § 6, exerc. 12) (in particular if K is a division ring with a valuation), the topology $\mathscr{T}_0$ is compatible with the vector space structure of E. If, further, K is not discrete, then E is not a Baire space for any topology that is finer than $\mathscr{T}_0$ and compatible with the vector space structure of E. b) Moreover, if there does not exist in K any neighbourhood of 0 bounded on the right (see c)) give an example of a family $(E_\iota)$ such that the topology $\mathscr{T}_0$ is not compatible with the vector space structure of E. c) Let $A = \mathbf{R}[X]$ be the ring of polynomials in one variable on $\mathbf{R}$. For every sequence $s = (\varepsilon_n)_{n \geq 0}$ of real numbers $> 0$, denote by $V_s$ the set of polynomials $\sum_k a_k X^k \in A$ such that $|a_k| < \varepsilon_k$ for all k. Let $\mathfrak{T}$ be the set of the $V_s$ where s varies in the set of sequences of numbers $> 0$. Show that $\mathfrak{T}$ is a fundamental system of symmetric neighbourhoods of 0 for a topology compatible with the ring structure of A. Let $K = \mathbf{R}(X)$ be the division ring of fractions of A; denote by $\mathfrak{S}$ the family of subsets of K of the form $U(1 + U)^{-1}$, where U varies in the set of the $V_s$ not containing 1; show that $\mathfrak{S}$ is a fundamental system of neighbourhoods of 0 for a topology compatible with the division ring structure of K, and that there does not exist in K any neighbourhood of 0 that is bounded.

d) For every Hausdorff topological division ring K, show that there exists a set I such that on F = K^I, the topology $\mathscr{T}$, defined above, is not compatible with the vector space structure of F.