PARTIALLY ORDERED NORMED lINEAR SPACES WITH WEAK FATOU PROPERTY

Abstract

Let $E$ be a Riesz space with lattice ordered norm $\|\cdot\|$. Amemiya proved that $E$ is complete under this norm if $E$ has weak Fatou property for monotone sequence (: $E$ is monotone complete) with respect to the norm $\|\cdot\|$. This is a generalization of the Riesz-Fisher's and the Nakano's theorem. In the cases of non normed Riesz space or non lattice ordered norm, this theorem is not true in general. We shall investigate in this paper a necessary and sufficient condition for Amemiya's theorem to be valid in a partially ordered normed linear space.