Exercise 16, § 2

With the notations of exerc. 14, let $E''$ be the dual of $E'' = \ell^\infty(A)$. a) Let $e_\alpha$ (for $\alpha \in A$) be the element of $E''$ for which $e_\alpha(\beta) = \delta_{\alpha \beta}$ (Kronecker's symbol). Let $(K_n)$ be a sequence of finite subsets of A, two by two disjoint, and let $(x'''_n)$ be a sequence of points in $E'''$. Show that there exists $(n_k)$, a strictly increasing infinite sequence of integers $> 0$ such that, if we put $B = \bigcup_k K_{n_k}$, then the elements $y'_n = (x'''_n(\alpha))_{\alpha \in B}$ belong to $\ell^1(B)$. (Let $\delta$ be an arbitrary number $> 0$, and let $(J_m)_{m \in \mathbf{N}}$ be a partition of $\mathbf{N}$ in finite sets. Arguing by reductio ad absurdum, show that if $x''' \in E'''$, then there exists an integer m such that $|\langle x'', x''' \rangle| \leq \delta$ for all $x'' \in E''$ for which $\|x''\| \leq 1$ and $x''(\alpha) = 0$ except for indices $\alpha$ belonging to the set $\bigcup_{n \in J_m} K_n$. Apply this result successively to $x'''_1, x'''_2, ...$ in a suitable way.) b) Deduce from a) and from exerc. 15, a) that, if $(x'''_n)$ is a sequence converging to 0 in $E'''$ for the topology $\sigma(E''' , E'')$ and if $\tilde{x'''_n}$ is the restriction of $x'''_n$ to the strongly closed subspace E of $E''$, then $\lim_{n \to \infty} \| \tilde{x'''_n} \| = 0$ in $E'$.

c) Deduce from b) that, the strongly closed subspace E of E'' does not have a topological complement for the strong topology. (Restricting to the case A = N, let $(e'_n)$ be the sequence of continuous linear forms on E such that $\langle x, e'_n \rangle = x(n)$ for all $x \in E$; show that the sequence $(e'_n)$ tends to 0 for $\sigma(E', E)$, but that $e'_n$ cannot be extended to a continuous linear form $x'''_n$ on $E''$ in such a way that the sequence $(x'''_n)$ tends to 0 for $\sigma(E'''', E'')$.)