Exercise 20, § 6

Let $F = \mathbf{R}^{(A)}, G = \mathbf{R}^A$, where A is any infinite set; suppose that F and G are put in separating duality by the bilinear form $\langle x, y \rangle = \sum_{\alpha \in A} x(\alpha) \, y(\alpha)$. a) Let N be an additive subgroup of G; we denote by $N^*$ the subgroup of the $x \in F$ such that $\langle x, y \rangle$ is an integer for all $y \in N$ and by $N^{**}$ the subgroup of the $z \in G$ such that $\langle x, z \rangle$ is an integer for all $x \in N^*$. If $\overline{N}$ is the closure of N for the topology $\sigma(G, F)$, show that $N^*$ is closed in F for $\sigma(F, G)$ and that $N^{**} = \overline{N}$ (to establish this last point, use GT, VII, § 1.3, prop. 6, projecting N on the finite dimensional coordinate varieties of G). b) Suppose that $A = N$. Let M be a closed subgroup of F for $\sigma(F, G)$; show that if V is the largest vector subspace contained in M, then M is the topological direct sum of V and of a closed subgroup P that is a free $\mathbf{Z}$-module having an enumerable base. (Consider F as the union of an increasing sequence $(F_n)$ of finite dimensional vector subspace and apply GT, VII, § 1.2, th. 2 and § 1, exerc. 7.) P is discrete (for the topology induced by $\sigma(F, G)$), if, and only if, P is of finite rank. c) Deduce from a) and b) that when $A = N$, every closed subgroup of G (when G carries the product topology $\sigma(G, F)$) can be transformed, by an automorphism of the topological group G, in a product $\mathbf{R}^I \times \mathbf{Z}^J$, where I and J are two sets of N without common elements. d) In the space $E = \mathbf{R}^N$, carrying the topology $\sigma(E, E^*)$, show that the subgroup $\mathbf{Z}^N$ is closed and does not contain any line, even though it is not a free $\mathbf{Z}$-module (A, VII, p. 59, exerc. 8); the results of b) do not therefore extend when A is not enumerable.