Proposition 13

In the Riesz space $L^p$, equipped with the topology defined by the norm $\| \tilde{f} \|_p$, the mapping $\tilde{f} \mapsto | \tilde{f} |$ is uniformly continuous, and the set of elements $\tilde{f} \geq 0$ is closed.

The first part of the proposition follows at once from Prop. 11 of No. 5; since the set of $\tilde{f} \geq 0$ is also the set of $\tilde{f}$ such that $| \tilde{f} | = f$, it is closed, because $\tilde{f} \mapsto | \tilde{f} |$ is a continuous mapping and $L^p$ is Hausdorff.

We thus see that the topology on $L^p$ defined by the norm $\| \tilde{f} \|_p$ is compatible with the ordered vector space structure of $L^p$ (TVS, II, § 2, No. 7).