Exercise 14, § 2

Let A be an infinite set. a) Show that the strong dual of the Banach space $E = \overline{\mathscr{K}(A)}$ (IV, p. 47, exerc. 1) can be identified with the Banach space $\ell^1(A)$, and that the strong dual of the Banach space $\ell'(A)$ can be identified with the Banach space $\mathscr{B}(A) = \ell^\infty(A)$; deduce that E is not reflexive and that $E''/E$ is infinite dimensional (cf. IV, p. 71, exerc. 18). If $A = \mathbf{N}$, then E and E' are Banach spaces which satisfy the first axiom of countability, but $E''$ does not (I, p. 25, exerc. 1). b) Let $B''$ be the unit ball in $E'' = \ell^\infty(A)$, and let $B''_0$ be the convex set $B'' + (B'' \cap E)$. Show that $B''_0$ is a bounded closed convex set in $E''$ for the strong topology, with a non-empty interior, but does not have any extremal point. If $p$ is the gauge of $B''_0$, show that $E''$ endowed with the norm $p$ is not isometric to any dual of a Banach space and that $B''_0$ is not closed for the topology $\sigma(E'', E')$ (although $B''$ is compact for $\sigma(E'', E')$ and $B'' \cap E$ is strongly closed).