Exercise 13, § 1

Let E be a vector space of infinite dimension over $\mathbf{R}$, and let $\mathfrak{T}$ be the family of all absorbent and balanced sets of E. Show that $\mathfrak{T}$ does not satisfy axiom (EV$_{\text{III}}$) (in other words is not a fundamental system of neighbourhoods of 0 for a topology compatible with the additive group structure of E). For this, consider an infinite independent family $(e_n)_{n \geq 1}$ in E; for every integer $n \geq 1$, let $A_n$ be the set of points $\sum_{i=1}^n t_i e_i$ such that $|t_i| \leq 1/n$ for $1 \leq i \leq n$; let A be the union of the $A_n$, and V be a subspace complementary to the subspace of E generated by the $e_n$, and write C for the set $A + V$; show that there exists no set $M \in \mathfrak{T}$ such that $M + M \subset C$.