Exercise 15, § 2

The notations are those of exerc. 14. a) Let $(x'_n)$ be a sequence in $E'$ which converges to 0 for the topology $\sigma(E', E'')$; show that, for every $\varepsilon > 0$, there exists a finite subset H of A such that $\sum_{\alpha \notin H} |x'_n(\alpha)| \leq \varepsilon$ for every integer n. (Argue by reductio ad absurdum : if the property were not true, show that then there exists a number $\delta > 0$, an increasing sequence $(n_k)$ of integers, an increasing sequence $(H_k)$ of finite subsets of A, such that $\sum_{\alpha \in H_{k-1}} |x'_n(\alpha)| \leq \frac{\delta}{8}$ for $n \geq n_k$, $\sum_{\alpha \notin H_k} |x'_n(\alpha)| \leq \frac{\delta}{8}$ for $n \leq n_k$ and $\sum_{\alpha \in H_k - H_{k-1}} |x'_n(\alpha)| \geq \frac{\delta}{2}$; show that this implies a contradiction (the « gliding bump » method).) Deduce that the sequence $(x'_n)$ converges to 0 for the strong topology, although the latter is strictly finer than the topology $\sigma(E', E'')$. b) Show that if $(x'_n)$ is a Cauchy sequence in $E'$ for the topology $\sigma(E', E'')$, it converges to a point in $E'$ for this topology ; in other words, $E'$ is semi-complete (III, p. 7) for $\sigma(E', E'')$. (Show that, for every $\varepsilon > 0$, there exists a finite subset H in A such that $\sum_{\alpha \notin H} |x'_n(\alpha)| \leq \varepsilon$ for every integer n ; argue by reductio ad absurdum, as in a), and use a).)