Exercise 17, § 4

For a subset A of a commutative group G, written additively, and for each $n > 0$ denote the set of elements of the form $\sum_{i=1}^n x_i$, where $x_i \in A$ for all i by $^n + A$. We say that the set A of G is convex if, for every integer $n > 0$ the relation $nx \in ^n + A$ implies $x \in A$. a) Show that if a commutative topological group G (written additively) is isomorphic to a subgroup of the additive group of a locally convex vector space (with the induced topology) then there exists a fundamental system of symmetric convex neighbourhoods of 0 in G. b) Conversely, let G be a Hausdorff topological commutative group (written additively) in which there exists a fundamental system, $\mathfrak{B}$, of symmetric convex neighbourhoods of 0. Show that G is without torsion, and, hence, can be considered (algebraically) as a subgroup of the additive group of a vector space on the field $\mathbf{Q}$ (A, II, § 7.10, cor. 1 to prop. 26). For every set $V \in \mathfrak{B}$, let $\tilde{V}$ be the set of elements $rx$ where $x \in V$ and r varies in the set of rational numbers such that $0 \leq r \leq 1$; show that $\tilde{V}$ is symmetric and convex (in the sense defined above). Deduce, further, that if there is no open subgroup of G distinct from G itself, then the sets $\tilde{V}$ form a fundamental system of neighbourhoods of 0 for a topology compatible with the vector space structure of E on $\mathbf{Q}$ ($\mathbf{Q}$ being given its usual topology); conclude that in this case G is isomorphic to a subgroup of the additive group of a Hausdorff locally convex space. c) Let G be the group $\mathbf{R} \times \mathbf{R}$ ordered lexicographically (A, VI, p. 7); consider the Hausdorff topology $\mathscr{T}_0(G)$ on G that is compatible with its group structure (GT, IV, § 1, exerc. 1). Show that for this topology there exists a fundamental system of symmetric convex neighbourhoods of 0, but that G is not isomorphic to any subgroup of the additive group of a Hausdorff topological vector space over $\mathbf{R}$.