Exercise 21, § 2

Let E be a Banach space, E' its dual, B the unit ball in E and $B'_r$ the closed ball with centre 0 and radius r in E'. a) Let $M'_1, M'_2$ be two vector subspaces of E', which are everywhere dense for the weak topology $\sigma(E', E)$. In order that the topologies induced by $\sigma(E, M'_1)$ and $\sigma(E, M'_2)$ on B coincide, it is necessary and sufficient that the strong closures of $M'_1$ and $M'_2$ in E' are identical. b) Let M' be a vector subspace of E' which is everywhere dense for $\sigma(E', E)$; let ${M'}^{(1)}$ denote the vector subspace generated by the closure of $M' \cap B'_1$ in E' for the topology $\sigma(E', E)$. In order that ${M'}^{(1)} = E'$, it is necessary and sufficient that the weak closure of $M' \cap B'_1$ contains a ball $B'_r$ with $r > 0$ (use the fact that E' is barrelled for the strong topology). c) Let r be the supremum of the numbers t such that the weak closure of $M' \cap B'_1$ contains a ball $B'_t$; the number r is said to be the characteristic of M'. Show that r is the infimum of the numbers $\sup_{x' \in M' \cap B'_1} |\langle x, x' \rangle| / \|x\|$ where x ranges over the set of all points $\neq 0$ in E (use the Hahn-Banach theorem). d) Show that $1/r$ is the supremum of $\|x\|$ as x ranges over the closure of B in E for the topology $\sigma(E, M')$ (use c) and the Hahn-Banach theorem).

e) Let ${M'}^\circ$ be the orthogonal of $M'$ in $E''$; show that $r = \inf(\|x + z''\|/\|x\|)$ where $z''$ ranges over ${M'}^\circ$ and $x$ ranges over the set of non-zero points in $E$ (use c) and the Hahn-Banach th). Deduce that in order that ${M''}^{(1)} = E'$, it is necessary and sufficient that $E + {M'}^\circ$ be strongly closed in $E''$ (use Banach’s th., I, p. 17).

f) Let $A = \mathbf{N} \times \mathbf{N}$ and $E = \mathscr{K}(A)$ (cf. IV, p. 54, exerc. 14). In the space $E'' = \ell^\infty(A)$, let $P$ be the vector subspace consisting of all points $x = (x_{ij})$ such that $x_{ij} = x_{0j}/(j+1)$ for all $i \geq 0$. Show that $P = {M'}^\circ$ where $M'$ is a vector subspace of $E'$ that is everywhere dense (for $\sigma(E', E)$), but that $E + {M'}^\circ = E + P$ is not strongly closed in $E''$; deduce that the characteristic of $M'$ is 0.