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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: Ting-On To
Journal: Proc. Amer. Math. Soc. 30 (1971), 287-295
MSC: Primary 46A40
MathSciNet review: 0417746
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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.

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Keywords: Least upper bound property, Hahn-Banach extension property, lineally closed wedges, semispace-wedges, lexicographically ordered linear spaces
Article copyright: © Copyright 1971 American Mathematical Society