Exercise 29, § 1

Let $E$ be a normed space, $E'$ its dual. We say that a mapping $f$ of $E - \{0\}$ into $E' - \{0\}$ is a support mapping if it is positively homogeneous, and if for every $x \in E$ such that $\|x\| = 1$, one has $\|f(x)\| = 1$ and $\langle x, f(x) \rangle = 1$. In order that $E$ be smooth (V, p. 65, exerc. 26), it is necessary and sufficient that there exist a unique support mapping of $E - \{0\}$ 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 smoothness point 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 provided with the topology of the norm, and $S'$ with the weak topology $\sigma(E', E)$. γ) For every $y \in E$, the mapping $t \mapsto \|x_0 + ty\|$ admits a derivative at the point $t = 0$. (To see that α) implies β), reason by contradiction using the weak compactness of the unit ball in $E'$. To see that β) implies γ), reduce to the case where $E$ is of dimension 2 and use the fact that $t \mapsto \|x_0 + ty\|$ is convex.) Every support mapping is then continuous at the point $x_0$.