Exercise 10, § 2

Let K be a compact space. For every Hausdorff quotient space $K_1$ of K, one identifies the Banach space $\mathscr{C}(K_1)$ with a closed subspace of $\mathscr{C}(K)$ by means of the injection $f \mapsto f \circ \pi$, where $\pi$ is the canonical mapping of K onto $K_1$.

a) Let u be a continuous linear mapping of $\mathscr{C}(K)$ into a quasi-complete Hausdorff locally convex space F. In order that u transform the unit ball of $\mathscr{C}(K)$ into a subset of F that is relatively compact for $\sigma(F, F')$, it suffices that, for every metrizable quotient space $K_1$ of K, the restriction of u to $\mathscr{C}(K_1)$ have that property. (Make use of Eberlein’s theorem (TVS, IV, §5, No. 3, Th. 1), and observe that every sequence $(f_n)$ in $\mathscr{C}(K)$ is contained in a subspace $\mathscr{C}(K_1)$, where $K_1$ is a metrizable quotient space of K; for that, one considers the continuous mapping $x \mapsto (f_n(x))$ of K into $\mathbf{R}^N$.)

b) For every Hausdorff quotient space $K_1$ of K, the transpose of the canonical injection of $\mathscr{C}(K_1)$ into $\mathscr{C}(K)$ is the mapping $\mu \mapsto \pi(\mu)$ of $\mathscr{M}(K)$ into $\mathscr{M}(K_1)$. Denote

²This lemma, suppressed from the second edition of Ch. V, asserts that $\int^* fg\, d\mu = \inf \int^* \varphi g\, d\mu$, where f and g are numerical functions $\geq 0$, g is measurable, and $\varphi$ runs over the set of measurable functions $\geq f$.

by $(\mathscr{M}(K))'$ the dual of the Banach space $\mathscr{M}(K)$ (the bidual of $\mathscr{C}(K)$). For a subset A of $\mathscr{M}(K)$ to be relatively compact for $\sigma(\mathscr{M}(K), (\mathscr{M}(K))')$, it is necessary and sufficient that, for every metrizable quotient space $K_1$ of K, the canonical image of A in $\mathscr{M}(K_1)$ be relatively compact for $\sigma(\mathscr{M}(K_1), (\mathscr{M}(K_1))')$. (Observe that A is strongly bounded; one can then suppose that A is convex, balanced and closed for the vague topology $\sigma(\mathscr{M}(K), \mathscr{C}(K))$ (apply in $\mathscr{M}(K_1)$ Exer. 10 of Ch. IV, §7); A is therefore the image of the unit ball of the dual $F'$ of a Banach space F, under the transpose of a continuous linear mapping $u$ of $\mathscr{C}(K)$ into F (consider the polar of A in $\mathscr{C}(K)$); then apply a).)