Exercise 30, § 1

Let E be 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 non-reflexive ; then there exists $x' \in E'$ such that $\|x'\| = 1$, but that we do not have $\langle x, x' \rangle = 1$ for any $x \in E$ with $\|x\| = 1$ (IV, p. 57, exerc. 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 all $n$ (II, p. 77, exerc. 4). Show that in E'', the sequence $(x_n)$ does not converge to any point for the topology $\sigma(E'', E''')$ (observe that otherwise it would converge to a point $x \in E$ for which $\langle x, x' \rangle = 1$). b) Show that it is not possible that $x'$ and $x'_n$ are points of smoothness of the unit sphere in E''' (observe that $x_n$, considered as an element of $E^{IV}$ will 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 exerc. 29). c) Conclude that if E''' is smooth, or if E^{IV} is strictly convex, then E is necessarily reflexive.