Exercise 29, § 1

Let E be a normed space, E' its dual. A mapping $f$ from $E - \{0\}$ into $E' - \{0\}$ is said to be a support mapping if it is positively homogeneous, and if for every $x \in E$ such that $\|x\| = 1$, we have $\|f(x)\| = 1$ and $\langle x, f(x) \rangle = 1$. For E to be smooth (V, p. 66, exerc. 26), it is necessary and sufficient that there exists a unique support mapping from $E - \{0\}$ into $E' - \{0\}$.

Let S be the unit sphere in E, S' the unit sphere in E', and let $x_0 \in S$. The following conditions are equivalent : α) $x_0$ is a point of smoothness of the unit ball in E, β) There exists a support mapping $f$ whose restriction to S is continuous at the point $x_0$ when S is assigned the norm topology, and S' the weak topology $\sigma(E', E)$. γ) For every $y \in E$, the mapping $t \mapsto \|x_0 + ty\|$ has a derivative at the point $t = 0$. (To see that α) implies β), argue by reductio ad absurdum, using the weak compactness of the unit ball in E'. To see that β) implies γ), reduce to the case where E is 2-dimensional and use the fact that $t \mapsto \|x_0 + ty\|$ is convex.) Then every support mapping is continuous at the point $x_0$.