Exercise 34, § 1

Let E be a Banach space and K be a convex, bounded subset that is closed for $\sigma(E, E')$. Suppose that for every sequence $(x_n)$ in K, there exists an extracted sequence $(x_{n_k})$ such that the sequence of averages $(x_{n_1} + \cdots + x_{n_k})/k$ is convergent for $\sigma(E, E')$. Show 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 hypothesis 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 has the property of exerc. 33, d), then E is a reflexive space (cf. IV, p. 57, exerc. 25).