Exercise 12, § 2

Let E be a Banach space and E' its dual. a) Show that the distance $x \mapsto d(x, A)$ from a point $x \in E$ to a closed convex set A is a lower semi-continuous function on E for the topology $\sigma(E, E')$. b) Show that if E is reflexive, then for every closed convex subset A of E, there exists a point $x_0 \in A$ such that $\|x_0\|$ is equal to the distance of 0 to A (use a)). This point is unique if every boundary point of the unit ball of E is extremal (II, p. 54, def. 1). c) Suppose E is reflexive and let B be a closed convex and bounded subset in E; deduce from a) and b) that there exist two points $x \in A, y \in B$ such that $\|x - y\| = d(A, B)$ (cf. V, p. 71, exerc. 8).