Exercise 16, § 4

a) Let E be a vector space with an enumerable basis. Show that the finest locally convex topology on E is the finest of the topologies on E (compatible or not with the vector space structure of E) which induces the canonical topology on every finite dimensional subspace of E. b) Let $E_0$ be an infinite dimensional Banach space. Let E be the vector space that is the direct sum of $E_0$ of $\mathbf{R}^{(N)}$ and let $E_p$ be the subspace of E that is the direct sum of $E_0$ and of $\mathbf{R}^p$ (identified as the product of the first p factors of $\mathbf{R}^{(N)}$; we give to $E_p$ the product topology of those of its factor, so that the topology of $E_p$ is induced by that of $E_{p+1}$. Show that on E the inductive limit topology of those of the $E_p$ is not the finest of the topologies (compatible or not with the vector-space structure of E) which induces on each $E_p$ a coarser topology than that of $E_p$. We can proceed as follows : $\alpha)$ Let q be a norm on $E_0$ which defines a topology strictly coarser than that of $E_0$ (II, p. 74, exerc. 5). For every $\varepsilon > 0$ define a mapping $f_\varepsilon$ of $E_0$ in $\mathbf{R}_+$ by the relation $f_\varepsilon(x) = \sup(q(x), \varepsilon - \|x\|)$. Show that $f_\varepsilon$ is continuous and $> 0$ in $E_0$ and that $\inf_{\|x\| = \varepsilon} f_\varepsilon(x) = 0$. $\beta)$ Let U be the subset of E formed by the $(x, (t_n))$ such that $t_n < f_{1/n}(x)$ for all n. Show that $U \cap E_p$ is open in $E_p$ for all p. $\gamma)$ Show that if $V \subset U$ is an absorbent convex set, then $V \cap E_0$ cannot contain any ball with centre 0 in $E_0$.