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

DOI:
https://doi.org/10.1090/S0002-9939-1971-0417746-3

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 , whose induced wedge satisfies and {zero vector}, then *V* fails to have the Hahn-Banach extension property. From this the desired result follows.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0417746-3

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