Exercise 26, § 1

One says that a real normed space $E$ is smooth if all the points of the unit sphere in $E$ are points of smoothness (II, p. 93, Exercise 6) of the unit ball. For this to be so, it is necessary and sufficient that there exist one and only one mapping $f$ from $E - \{0\}$ into $E' - \{0\}$, positively homogeneous, such that $\|f(x)\| = 1$ for $\|x\| = 1$, and that $\langle x, f(x) \rangle = \|x\| \cdot \|f(x)\|$. Now prove that the following properties are equivalent: α) $E$ is smooth. β) For every $x \neq 0$ in $E$ and every $y \in E$, there exists a unique scalar $\alpha$ such that $\alpha x + y$ is quasi-normal to $x$.

γ) For every $x \in E$, if $y$ and $z$ are quasi-normal to $x$, $y + z$ is quasi-normal to $x$. (To see that γ) implies β), observe that if $\alpha x + y$ and $\beta x + y$ are quasi-normal to $x$, $(\alpha - \beta) x$ is quasi-normal to $x$.)