Exercise 28, § 1

Let $E$ be a normed space, $E'$ its dual. a) Now prove that if $E'$ is smooth (V, p. 65, exer. 26), $E$ is strictly convex (if $x, y$ are such that $x \neq y$, $\|x\| = \|y\| = \|\frac{1}{2}(x + y)\| = 1$, consider an $x' \in E'$ such that one has $\|x'\| = 1$ and $\langle \frac{1}{2}(x + y), x' \rangle = 1$). b) Now prove that if $E'$ is strictly convex, $E$ is smooth.