Exercise 30, § 1

Soient $E$ a Banach space, $E'$ its strong dual, $E''$ the strong dual of $E'$, $E'''$ the strong dual of $E''$, $E^{IV}$ the strong dual of $E'''$. a) Suppose that $E$ is not reflexive; there then exists $x' \in E'$ such that $\|x'\| = 1$, but such that for no $x \in E$ such that $\|x\| = 1$ does one have $\langle x, x' \rangle = 1$ (IV, p. 57, Exercise 25). On the other hand, there exists a sequence $(x'_n)$ of points of $E'$ such that $\|x'_n\| = 1$, tending strongly to $x'$, and a sequence $(x_n)$ of points of $E$ such that $\|x_n\| = 1$ and $\langle x_n, x'_n \rangle = 1$ for every $n$ (II, p. 82, Exercise 4). Now prove that in $E''$, the sequence $(x_n)$ is convergent to no point for the topology $\sigma(E'', E''')$ (remark that otherwise it would converge to a point $x \in E$ and prove that one would have $\langle x, x' \rangle = 1$). b) Now prove that it is not possible that $x'$ and $x'_n$ are points of smoothness of the unit sphere in $E'''$ (remark that $x_n$, considered as an element of $E^{IV}$, would be the unique element $x_n^{IV} \in E^{IV}$ such that $\|x_n^{IV}\| = 1$ and $\langle x'_n, x_n^{IV} \rangle = 1$, and use Exercise 29). c) Conclude that if $E'''$ is smooth, or if $E^{IV}$ is strictly convex, $E$ is necessarily reflexive.