Exercise 4, § 2

Let E be a fully lattice-ordered space.

a) Let $L$ be a positive linear form on E, continuous for the topology $\mathscr{T}_0(E)$ (§1, Exer. 10). Show that the set of $x \in E$ such that $L(|x|) = 0$ is a band $Z(L)$. Let $S(L)$ be the band supplementary to $Z(L)$ in E (§1, Exer. 6).

b) Let $L$ and $M$ be two positive linear forms on E, continuous for $\mathscr{T}_0(E)$. In order that, in the space $\Omega$ of relatively bounded linear forms on E, $M$ belong to the band generated by $L$, it is necessary and sufficient that $S(M) \subset S(L)$. (To see that the condition is sufficient, argue by contradiction, on supposing that $M$ does not verify the condition of Prop. 5; consider a sequence $(y_n)$ of elements of E such that $0 \leq y_n \leq x$,

$L(y_n) \leq 1/2^n$ and $M(y_n) \geq \alpha > 0$, and infer the existence of an element $z \geq 0$ in E such that $L(z) = 0$ and $M(z) \geq \alpha .$

c) Let $L$ and $M$ be two positive linear forms on E, continuous for $\mathscr{T}_0(E)$. In order that, in $\Omega$, $L$ and $M$ be alien, it is necessary and sufficient that $S(L) \cap S(M) = \{0\}$. (To see that the condition is necessary, prove that if $S(L) \cap S(M)$ does not reduce to 0 then the bands in $\Omega$ generated by $L$ and $M$ have an element $\neq 0$ in common, by using b.)