Exercise 23, § 2

Let $T$ be a locally compact space, $\mu$ a positive measure on $T$, $F$ one of the three Banach spaces $\overline{\mathscr{K}(T)}$, $L^1(\mu)$, $L^\infty(\mu)$. Let $u$ be a continuous linear mapping from F into a quasi-complete space G, which transforms the unit ball of F into a relatively compact subset of G for $\sigma(G, G')$. Now prove that u transforms every relatively compact subset of F for $\sigma(F, F')$ into a relatively compact subset of G for the initial topology. (Apply exerc. 22 c) to reduce the case $F = L^1(\mu)$ to the case $F = L^\infty(\mu)$; use exerc. 13 of chap. II, § 1 to reduce the case $F = L^\infty(\mu)$ to the case $F = C(S)$ where S is a suitable compact space. If $F = \overline{\mathscr{K}(T)}$, apply exerc. 22 b), then use exerc. 24 b) and 17 of chap. V, § 5; it will be noted that a Cauchy sequence for $\sigma(F, F')$ converges simply in T, and fortiori converges in measure for every bounded measure on T.)