Exercise 2, § 2

In the product topological vector space E = $\mathbf{R}^N$ over the field $\mathbf{R}$, denote by $e_n$ ($n \in \mathbf{N}$) the elements of the canonical basis of the direct sum $\mathbf{R}^{(N)}$. Write $a_0 = e_0$, $a_n = e_0 + (1/n)e_n$ for $n \geq 1$. Show that, for every integer $n \geq 0$, the $a_i$ such that $0 \leq i \leq n$ form a topologically independent family in E, but that the infinite family $(a_n)_{n \geq 0}$ is not topologically independent. If M is the closed vector subspace $Ra_0$, the classes $\dot{a}_n$ of the $a_n$ in E/M form a topologically independent family (for $n \geq 1$), but the closed vector subspace N generated by the $a_n$, with index $n \geq 1$, in E contains M.