Exercise 17, § 1

Let A be the set of increasing mappings and $\geq 1$ from $\mathbf{N}$ into $\mathbf{N}$; for every $\alpha \in A$, denote by $B_\alpha$ the set of points $z = (z_n) \in \mathbf{R}^N$ such that $|z_n| \leq \alpha(n)$ for every $n \in \mathbf{N}$. a) Now prove that the sets $B_\alpha$ form a basis of the bornology of all bounded sets of the space $\mathbf{R}^N$. b) For each $\alpha \in A$, the set $RB_\alpha$ is a vector subspace of $\mathbf{R}^N$, distinct from $\mathbf{R}^N$ and dense in $\mathbf{R}^N$; there therefore exists a linear form $f_\alpha \neq 0$ (not continuous) on $\mathbf{R}^N$ such that $f_\alpha(z) = 0$ for every $z \in B_\alpha$. c) Let E be the vector space of mappings $g : \alpha \mapsto (g_n(\alpha)) \in \mathbf{R}^N$ from A into $\mathbf{R}^N$ such that for every $n \in \mathbf{N}$, the sum $p_n(g) = \sum_\alpha |g_n(\alpha)|$ is finite. Now prove that the $p_n$ are semi-norms which define on E a Fréchet space topology. d) Let H be the set of $h \in E$ such that $h(\alpha) \in RB_\alpha$ for every $\alpha \in A$; now prove that H is a vector subspace everywhere dense in E (remark 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 formed by the $g \in E$ such that $\sum_\alpha |f_\alpha(g(\alpha))| < +\infty$; the mapping $u : g \mapsto (f_\alpha(g(\alpha)))_{\alpha \in A}$ is therefore a linear mapping from $E_0$ into the Banach space $F = \ell^1(A)$ (I, p. 4). Now 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 every $\alpha \in A - J$ and that the $f_\alpha(g(\alpha))$ for $\alpha \in J$ take arbitrary values in $\mathbf{R}$). Now prove that $u^{-1}(0)$ is everywhere dense in $E_0$ (use d)). Finally, now prove that for every bounded subset C of E, there exists $\alpha_0 \in A$ such that $f_{\alpha_0}(g(\alpha_0)) = 0$ for every $g \in C \cap E_0$, 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 $E_0 \times F$, vector subspace of the Fréchet space $E \times F$. Now prove that G is everywhere dense in $E \times F$ (note that for every $x \in E_0,\ x + u^{-1}(0)$ is dense in E). However, now prove 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 $N = pr_1(M)$, remark that by virtue of e), $u(N)$ cannot contain U).