Exercise 17, § 1

Let A be the set of increasing mappings $\geq 1$ from $\mathbf{N}$ into $\mathbf{N}$; for every $\alpha \in A$, let $B_\alpha$ denote the set of all points $z = (z_n) \in \mathbf{R}^\mathbf{N}$ such that $|z_n| \leq \alpha(n)$ for all $n \in \mathbf{N}$. a) Show that the sets $B_\alpha$ form a base for the bornology of all bounded subsets of the space $\mathbf{R}^\mathbf{N}$. b) For every $\alpha \in A$, the set $\mathbf{R}B_\alpha$ is a vector subspace of $\mathbf{R}^\mathbf{N}$, distinct from $\mathbf{R}^\mathbf{N}$ and dense in $\mathbf{R}^\mathbf{N}$; hence there exists a linear form $f_\alpha \neq 0$ (not continuous) on $\mathbf{R}^\mathbf{N}$ such that $f_\alpha(z) = 0$ for all $z \in B_\alpha$. c) Let E be the vector space consisting of all mappings $g : \alpha \mapsto (g_n(\alpha)) \in \mathbf{R}^\mathbf{N}$ from A into $\mathbf{R}^\mathbf{N}$ such that for all $n \in \mathbf{N}$, the sum $p_n(g) = \sum_\alpha |g_n(\alpha)|$ is finite. Show that the $p_n$ are semi-norms which define the topology of a Fréchet space on E. d) Let H be the set of all $h \in E$ such that $h(\alpha) \in \mathbf{R}B_\alpha$ for all $\alpha \in A$; show that H is an everywhere dense vector subspace of E (observe that every $h \in E$ such that $h(\alpha) = 0$ except for a finite number of values of $\alpha \in A$ belongs to H). e) Let $E_0 \subset E$ be the vector subspace of E consisting of all $g \in E$ such that $\sum_\alpha |f_\alpha(g(\alpha))| < +\infty$; the mapping $u : g \mapsto (f_\alpha(g(\alpha)))_{\alpha \in A}$ is then a linear mapping from $E_0$ into the Banach space $F = \ell^1(A)$ (I, p. 4). Prove that $u(E_0)$ is everywhere dense in F (observe that for every finite subset J of A, there exists $g \in E_0$ such that $g(\alpha) = 0$ for all $\alpha \in A - J$ and that the $f_\alpha(g(\alpha))$ for $\alpha \in J$ take arbitrary values in $\mathbf{R}$). Show that $u^{-1}(0)$ is everywhere dense in $E_0$ (use d)). Finally, show that for every bounded subset C of E, there exists $\alpha_0 \in A$ such that $f_{\alpha_0}(g(\alpha_0)) = 0$ for all $g \in C \cap E_0$, and deduce that the closure of $u(C \cap E_0)$ in F is not a neighbourhood of 0 in F. f) Let G be the graph of u in $E_0 \times F$, a vector subspace of the Fréchet space $E \times F$. Show that G is everywhere dense in $E \times F$ (observe that for every $x \in E_0$, $x + u^{-1}(0)$ is dense in E). However, show that for every bounded subset M of G, the closure $\overline{M}$ of M in $E \times F$ does not contain the bounded set $\{0\} \times U$ of $E \times F$, where U is the unit ball in F (if $N = pr_1(M)$, observe that because of e), $u(N)$ cannot contain U).