Exercise 34, § 1

Let $E$ be a Banach space, $K$ a bounded convex set in $E$ and closed for $\sigma(E, E')$. Suppose that, for every sequence $(x_n)$ in $K$, there exists a subsequence $(x_{n_k})$ such that the sequence of means $(x_{n_1} + \cdots + x_{n_k})/k$ is convergent for $\sigma(E, E')$. Now prove that for every continuous linear form $x'$ on E, there exists an element $x$ of K such that $\langle x, x' \rangle = \sup_{y \in K} \langle y, x' \rangle$ (apply the assumption to a sequence $(x_n)$ of points of K such that $\langle x_n, x' \rangle$ tends to $\sup_{y \in K} \langle y, x' \rangle$).

Deduce that if E possesses the property of Exercise 33, d), it is a reflexive space (cf. IV, p. 57, Exercise 25).