Exercise 14, § 7

Let E be a closed vector subspace of the Banach space $l^\infty(\mathbf{N})$, formed of the sequences $x = (\xi_n)$ such that $\lim_{n \to \infty} \xi_n = 0$. a) Show that, in the Banach space E, the closed unit ball B does not have any extremal points. b) Let u be the continuous linear form $(\xi_n) \mapsto \sum_n 2^{-n} \xi_n$ on E. Show that there does not exist any support hyperplane of B that is parallel to the closed hyperplane with the equation $u(x) = 0$.