On positive total sets (Q1377382)

A subset \(A\) of the positive cone of the dual \(E'\) of a normed vector lattice \(E\) is called a positive total set if, for every \(x\in E\) it follows that \(x\) is positive, whenever \(\phi(x)\geq 0\) for all \(\phi\in A\). It is shown that the positive cone of a \(\sigma(E',E)\)-dense ideal of \(E'\) is positive total. An example of a normed vector lattice \(E\), inside \(c_0\), and a \(\sigma(E',E)\)-dense vector sublattice of \(E'\), that is not positive total, is provided as well.