Exercise 3, § 2

Let E be a fully lattice-ordered space.

a) Show that every linear form $L$ on E that is continuous for the topology $\mathscr{T}_0(E)$ (§1, Exer. 10) is relatively bounded, and that $|L|$ is continuous for $\mathscr{T}_0(E)$ (argue by contradiction, on observing that for every $x \in E$ the sequence of elements $x/n$ tends to 0 as $n$ tends to infinity, for the topology $\mathscr{T}_0(E)$).

b) Let A be any infinite set, $\mathscr{U}$ an ultrafilter on A whose sets have empty intersection. On the fully lattice-ordered space $E = \mathscr{B}(A)$, consider the positive linear form $x \mapsto \lim_{\mathscr{U}} x(t)$; show that this linear form is not continuous for the topology $\mathscr{T}_0(E)$.