The Equivalence of the Least Upper Bound Property and the Hahn-Banach Extension Property in Ordered Linear Spaces

Let V be a partially ordered (real) linear space with the positive wedge C. It is known that V has the least upper bound property if and only if V has the Hahn-Banach extension property and C is lineally closed. In recent papers, W. E. Bonnice and R. J. Silverman proved that the Hahn-Banach extension and the least upper bound properties are equivalent. We found that their proof is valid only for a restricted class of partially ordered linear spaces. In the present paper, we supply a proof for the general case. We prove that if V contains a partially ordered linear subspace W of dimension , whose induced wedge <!-- MATH $K = W \cap C$ --> satisfies <!-- MATH $K \cup ( - K) = W$ --> and <!-- MATH $K \cap ( - K) =$ --> {zero vector}, then V fails to have the Hahn-Banach extension property. From this the desired result follows.