Exercise 12, § 5

Let $E$ be a non reflexive Banach space. Prove that there exists a closed, non reflexive vector subspace $M$ of infinite codimension in $E$. (Let $(x_n)$ be a bounded sequence in $E$ which has no limit point for the topology $\sigma(E, E')$ (IV, p. 68, exerc. 8); by induction construct a sequence $(x_{n_k})$ extracted from $(x_n)$ and a topologically free sequence $(y_k)$ such that $\| x_{n_k} - y_k \| \leq 1/k$ for $k \geq 1$, and consider the closed vector subspace of $E$ generated by the $y_{2k}$.)