On the scalarization method in cone metric spaces

Recently, Du (J Nonlinear Anal 72:2259–2261, 2010) by using a nonlinear scalarization function, in the setting of locally convex topological vector spaces, could transfer a cone metric space to a usual metric space. Simultaneously, Amini-Harandi and Fakhar (Com Math Appl 59:3529–3534, 2010) by using a notion of base for the cone , in the setting of Banach spaces, could do the same. In this note we will see that two methods coincide and moreover they are valid for topological vector spaces and it is not necessary that we only consider the cones which have a compact base. Finally, it is worth noting that the nature of this note is similar to Caglar and Ercan (Order-unit-metric spaces, arXiv:1305.6070 [math.FA], 2013).