Exercise 8, § 2

Let E be a Riesz space, $(U_\iota)$ a family of positive linear forms on E. We consider the topology $\mathscr{T}$ on E defined by the semi-norms $U_\iota(|x|)$.

a) Show that the subspace H of E formed by the $x$ such that $U_\iota(|x|) = 0$ for all $\iota$ is an isolated subspace of E ($§ 1$, Exer. 4); the Hausdorff space associated with E is the quotient space $E/H$; by passage to the quotient, the $U_\ell$ define positive linear forms $\dot{U}_\ell$ on $E/H$, and the quotient topology on $E/H$ is defined by the semi-norms $\dot{U}_\ell(|x|)$.

b) Show that in the space $E$ the mapping $x \mapsto |x|$ is uniformly continuous, and deduce therefrom that if the topology $\mathscr{T}$ is Hausdorff then it is compatible with the ordered vector space structure of $E$.

c) Assume that the topology $\mathscr{T}$ is Hausdorff and that $E$ is fully lattice-ordered; show that every band $B$ in $E$ is closed, and that if $B'$ is the band formed by the elements alien to every element of $B$, then $E$ is the topological direct sum of $B$ and $B'$.

d) Assume that the topology $\mathscr{T}$ is Hausdorff. Let $\widehat{E}$ be the completion of $E$; if $P$ is the set of elements $\geqslant 0$ of $E$, show that the closure $\overline{P}$ of $P$ in $\widehat{E}$ defines a Riesz space structure on $\widehat{E}$ (cf. §1, No. 2).

e) If $E$ is Hausdorff and complete for the topology $\mathscr{T}$, show that in order for a set $A \subset E$, directed for the relation $\leqslant$, to have a supremum in $E$, it is necessary and sufficient that, for every index $\ell$, $U_\ell(x)$ be bounded above in $A$; for every continuous, increasing numerical function $f$ on $E$, one then has $\sup_{x \in A} f(x) = f(\sup A)$. Deduce from this that $E$ is fully lattice-ordered.

f) Assume that $E$ is Hausdorff and complete for the topology $\mathscr{T}$. Show that if a filter $\mathfrak{F}$ on $E$ has a limit for the order structure of $E$ (§ 1, Exer. 9), then it converges to the same limit for the topology $\mathscr{T}$ (make use of e)).