The equivalence of the least upper bound property and the Hahn-Banach extension property in ordered linear spaces

To, Ting-On

doi:10.1090/S0002-9939-1971-0417746-3

The equivalence of the least upper bound property and the Hahn-Banach extension property in ordered linear spaces
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Proc. Amer. Math. Soc. 30 (1971), 287-295 Request permission

Abstract:

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 $\geqq 2$, whose induced wedge $K = W \cap C$ satisfies $K \cup ( - K) = W$ and $K \cap ( - K) =$ {zero vector}, then V fails to have the Hahn-Banach extension property. From this the desired result follows.

References

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