Exercise 27, § 1

Let E be a separable real Banach space, $G = E'$ its dual, $G'$ a subspace of E that is dense in E, distinct from E, and barreled (TVS, III, §4, Exer. 4). Show that if G is equipped with the topology $\sigma(G, G')$, and its dual $G' = \mathscr{L}(G; \mathbf{R})$ with the topology of compact convergence, then $G'$ is not quasi-complete, and there exist a sequence $(x_n)$ in $G'$ tending to 0 and a sequence $(\lambda_n)$ of numbers $> 0$ and with finite sum, such that the series with general term $\lambda_n x_n$ is not convergent in $G'$. (Observe first that every subset of G bounded for $\sigma(G, G')$ is bounded for the norm on G. Consider a point $a \in E$ not belonging to $G'$, and a sequence $(a_n)$ of points of $G'$ that converges to $a$ in E and is such that $\|a_{n+1} - a_n\| \leq 1/2^n$.)