Exercise 6, § 2

Let E be a Riesz space, $\Omega$ the fully lattice-ordered space of relatively bounded linear forms on E, and F a co-lattice subspace of $\Omega$ ($§ 1$, Exer. 3). For every $x \in E$, the mapping $x' \mapsto \langle x, x' \rangle$ of F into $\mathbf{R}$ is a relatively bounded linear form $u_x$ on F, and the mapping $x \mapsto u_x$ is an increasing linear mapping of E into the fully lattice-ordered space $\Omega'$ of relatively bounded linear forms on F. In order that $x \mapsto u_x$ be an isomorphism of E onto a co-lattice subspace of $\Omega'$, that is, for $u_x > 0$ to imply $x > 0$, and that $u_{\sup(x, y)} = \sup(u_x, u_y)$, it is necessary and sufficient that the following two conditions be verified: 1° for every $x > 0$ in E, there exists an $x' > 0$ in F such that $\langle x, x' \rangle > 0$; 2° for every pair of alien elements $y \geq 0, z \geq 0$ of E, for every number $\varepsilon > 0$ and for every $x' \geq 0$ in F, there exist two elements $y' \geq 0, z' \geq 0$ of F such that $x' = y' + z'$ and $\langle y, y' \rangle + \langle z, z' \rangle \leq \varepsilon$. (Observe that if the second condition is verified and if $v$ is a positive linear form on F such that $v \geq u_x$, then $v(x') \geq \langle x^+, x' \rangle - \varepsilon$ for every $x' \geq 0$ in F and every $\varepsilon > 0$.)

If the condition 2° is fulfilled but not condition 1°, then the set of $x \in E$ such that $\langle x, x' \rangle = 0$ for all $x' \in F$ is an isolated linear subspace H of E. By passage to the quotient, the mapping $x \mapsto u_x$ then defines an isomorphism of the Riesz space E/H onto a co-lattice subspace of $\Omega'$.

Show that when $F = \Omega$, the above condition 2° is always verified (make use of Exer. 2).