Completely linear functionals in partially ordered spaces

For an arbitrary normed space $X$ the set $(X^{**})^\pi$ in $X^{**}$ introduced. It is proved that if $X$ is a $KN$-lineal then $\overline{X}^*=(X^{**})^\pi$, where $\overline{X}^*$ is the Nakano dual to the Banach dual $X^*$. By the same token $\overline{X}^*$ is not in any way related with any partial ordering in the $KN$-lineal $X$.