Exercise 28, § 1

Let E be a normed space, E' its dual. a) Show that if E' is smooth (V, p. 66, exerc. 26), E is strictly convex (if $x$ and $y$ are such that $x \neq y$, $\|x\| = \|y\| = \|\frac{1}{2}(x + y)\| = 1$, consider an $x' \in E'$ such that $\|x'\| = 1$ and

b) Show that if E' is strictly convex, E is smooth.