Exercise 5, § 2

a) Let E be a Riesz space, H an isolated linear subspace of E ($§ 1$, Exer. 4). Let $\Omega$ be the space of relatively bounded linear forms on E and let $\Theta$ be the subspace of $\Omega$ consisting of the linear forms $L \in \Omega$ that are zero on H. Show that $\Theta$ is an isolated subspace of $\Omega$ and that $\Theta$ is a fully lattice-ordered space isomorphic to the space of relatively bounded linear forms on the Riesz space E/H.

b) For a linear mapping $u$ of E into a Riesz space F, the following conditions are equivalent:

$\alpha ) \ u(\sup(x, y)) = \sup(u(x), u(y));$ $\beta ) \ u(\inf(x, y)) = \inf(u(x), u(y));$ $\gamma ) \ u(x^+) = (u(x))^+;$ $\delta )$ the relation $\inf(x, y) = 0$ implies $\inf(u(x), u(y)) = 0$.

One then says that $u$ is a latticial linear mapping.

c) For a linear subspace H of E to be maximal in the set of isolated subspaces $\neq E$, it is necessary and sufficient that it be of the form $\mathrm{Ker}(f)$, where $f$ is a latticial linear form $\neq 0$. (Make use of GT, V, $§ 3$, Exer. 1.)

d) Give an example of a Riesz space not reduced to 0 that contains no maximal isolated subspace (cf. $§ 1$, Exer. 5 b)).