Exercise 16, § 2

a) Let $\Lambda$ be the set of increasing mappings $\lambda : \mathbf{N} \to \mathbf{R}_+^*$; for every integer $n \geq 0$, and every $\lambda \in \Lambda$, let $\phi_n(\lambda) = \lambda(n)$. Let E be the set of all mappings $x : \Lambda \to \mathbf{C}$ such that, for every $n \in \mathbf{N}$, we have $p_n(x) = (\sum_{\lambda \in \Lambda} |x(\lambda)|^2 \phi_n(\lambda))^{1/2} < + \infty$. Show that E is a vector space on which the $p_n$ are the semi-norms defining a reflexive Fréchet space structure. b) Let B be a bounded set in E, and let $\alpha_n = \sup_{x \in B} p_n(x)$; let $\lambda_0$ be an element of $\Lambda$ such that $\lim_{n \to \infty} \lambda_0(n)^{-1} \alpha_n^2 = 0$. Show that $x(\lambda_0) = 0$ for all $x \in B$, and hence that the set B is not total in E. c) Let $(U_n)$ be a countable fundamental system of convex and balanced neighbourhoods of 0 in E; if $U_n^\circ$ is metrizable for the strong topology on $E'$, then there exists a sequence $(B_{nm})_{m \geq 0}$ of bounded sets in E such that the sets $B_{nm}^\circ \cap U_n^\circ$ form a fundamental system of neighbourhoods of 0 in $U_n^\circ$ for the strong topology. Deduce from b) that there exists an integer $n$ such that $U_n^\circ$ is not metrizable for the strong topology (use exerc. 5 of III, p. 38).