Exercise 3, § 1

a) Let E be a Riesz space, H a linear subspace of E. If, for any pair of elements $x, y$ of H, $\sup(x, y)$ belongs to H, then H is said to be a co-lattice subspace. For this to be the case, it is necessary and sufficient that the relation $x \in H$ imply $|x| \in H$.

b) Let I be a compact interval in $\mathbf{R}$, H the subspace of $\mathbf{R}^I$ formed by the restrictions to I of the first-degree polynomials $t \mapsto \alpha t + \beta$. Show that H is not a co-lattice subspace of $\mathbf{R}^I$ but that H is a Riesz space (for the structure induced by that of $\mathbf{R}^I$).