Exercise 25, § 2

Let $E$ be a Banach space satisfying the first axiom of countability. Show that the following properties are equivalent : a) $E$ is not reflexive. b) For every number $\theta$ such that $0 < \theta < 1$, there exists a sequence $(x'_n)$ in $E'$ such that, $\|x'_n\| \leq 1$ for all $n$, that the sequence $(x'_n)$ converges to 0 for $\sigma(E', E)$ and that the distance of 0 from the convex set generated by the $x'_n$ is $\geq \theta$. c) For every number $\theta$ such that $0 < \theta < 1$ and every sequence $\lambda_n$ of numbers $> 0$ such that $\sum_n \lambda_n = 1$, there exists a number $\alpha$ such that $\theta \leq \alpha \leq 1$ and a sequence $(y'_n)$ of points of $E'$ such that $\|y'_n\| \leq 1$ for all $n$, $\|\sum_n \lambda_n y'_n\| = \alpha$ and $\left\| \sum_{k=1}^n \lambda_k y'_k \right\| \leq \alpha (1 - \theta \sum_{j=n+1}^\infty \lambda_j)$ for all $n$. d) There exists $z' \in E'$ such that for no $x \in E$ do we have $|\langle x, z' \rangle| = \|x\| \cdot \|z'\|$ (Theorem of James-Klee). (To see that a) implies b), observe that there exists $z'' \in E''$ such that $\|z''\| < 1$ and $d(z'', E) > \theta$. If $(x_n)$ is an everywhere dense sequence in $E$, find the sequence $(x'_n)$ in $E'$ such that $\|x'_n\| < 1$ and such that $\langle x_k, x'_n \rangle = 0$ for $k \leq n$ and $\langle x'_n, z'' \rangle = \theta$. To see that b) implies c), use exerc. 24. For c) implies d), show that for all $x \in E$ we have $\left| \sum_n \lambda_n \langle x, y'_n \rangle \right| < \alpha$, with the notations of c).