Exercise 23, § 2

Let T be a locally compact space, $\mu$ a positive measure on T, and F one of the three Banach spaces $\mathscr{K}(T)$, $L^1(\mu)$, $L^\infty(\mu)$. Let u be a continuous linear mapping of F into a quasi-complete space G, that transforms the unit ball of F into a subset of G that is relatively compact for $\sigma(G, G')$. Show that u transforms every subset of F relatively compact for $\sigma(F, F')$ into a subset of G relatively compact for the original topology. (Apply Exer. 22 c) to reduce the case $F = L^1(\mu)$ to the case $F = L^\infty(\mu)$; use Exer. 13 of Ch. II, §1 to reduce the case $L^\infty(\mu)$ to the case $F = \mathscr{C}(S)$, where S is a suitable compact space. If $F = \mathscr{K}(T)$, apply Exer. 22 b), then use Exers. 22 b) and 15 of Ch. V, §5; note that a Cauchy sequence for $\sigma(F, F')$ converges pointwise on T, and a fortiori converges in measure for every bounded measure on T.)