Exercise 26, § 1

A real normed space E is said to be smooth if all the points of the unit sphere in E are points of smoothness (II, p. 87, exerc. 6) of the unit ball. For this to be so, it is necessary and sufficient that there exists a unique positively homogeneous mapping $f$ from $E - \{0\}$ into $E' - \{0\}$, such that $\|f(x)\| = 1$ for $\|x\| = 1$, and that $\langle x, f(x) \rangle = \|x\| \cdot \|f(x)\|$. Show that the following properties are equivalent : $\alpha)$ E is smooth.

β) For all $x \neq 0$ in E and all $y \in E$, 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 $x$, then $(\alpha - \beta) x$ is quasi-normal to $x$.)