Exercise 6, § 6

Let $I$ be the interval $[1, 2]$ with the discrete topology. Let $E$ be the complete normed space of functions $f : I \to \mathbf{R}$ such that $\|f\| = \sum_{i \in I} |f(i)| < +\infty$. For all $x \in I$, let $\varepsilon_x$ be the element of $E$ such that $\varepsilon_x(t) = 0$ for $t \neq x$ and $\varepsilon_x(x) = 1$. Let $P$ be the subgroup of $E$ generated by the $x\varepsilon_x$ with $x \in I$; it is a discrete subgroup of $E$. Let $G$ be the Lie group $E/P$. Let $F$ be the hyperplane of $E$ consisting of the $f \in E$ such that $\sum_{i \in I} f(i) = 0$. Let $H$ be the Lie group $F/(F \cap P)$. Let $\phi$ be the canonical morphism of $H$ into $G$. Show that $\phi$ is bijective and is an immersion but is not a Lie group isomorphism. (Let $f$ be a linear form on $E$ with kernel $F$; show that $f(P) = \mathbf{R}$ and hence that $F + P = E$.) Deduce that in Proposition 3 the countability hypothesis cannot be omitted.