Example

Let E be a Riesz space equipped with a locally convex topology compatible with its ordered vector space structure (TVS, II, §2, No. 7). Let E' be the topological dual of E, and suppose in addition that the cone P of elements $\geqslant 0$ of E is complete for the weakened topology $\sigma(E, E')$. Then every continuous linear form $x' \in E'$ is relatively bounded, for one knows (TVS, II, §6, No. 8, Cor. 2 of Prop. 11) that under these conditions, for every $x \geqslant 0$ in E the set of $y \in E$ such that $|y| \leqslant x$ is compact for $\sigma(E, E')$. From this we deduce that E is then fully lattice-ordered; for ($§ 1$, No. 3, Prop. 2), it suffices to show that for every set $H \subset E$ that is bounded above and directed for $\leqslant$, the section filter $\mathscr{F}$ of H is convergent in E for the topology $\sigma(E, E')$ (the latter being compatible with the ordered vector space structure of E). By translation, we can suppose that $H \subset P$, and it then suffices to show that $\mathscr{F}$ is a Cauchy filter for $\sigma(E, E')$, or again that every continuous linear form $x' \in E'$ has a limit with respect to $\mathscr{F}$. But this follows at once from the monotone limit theorem when $x'$ is a positive linear form, and since every linear form $x' \in E'$ is the difference of two positive linear forms (Th. 1) our assertion is proved.

Exercises