Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1971-0417746-3
MathSciNet review: 0417746
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] W. E. Bonnice and R. J. Silverman, The Hahn-Banach extension and the least upper bound properties are equivalent, Proc. Amer. Math. Soc. 18 (1967), 843-849. MR 35 #5895. MR 0215050 (35:5895)
  • [2] -, The Hahn-Banach theorem for finite dimensional spaces, Trans. Amer. Math. Soc. 121 (1966), 210-222. MR 32 #2879. MR 0185412 (32:2879)
  • [3] T. O. To, A note of correction to a theorem of W. E. Bonnice and R. J. Silverman, Trans. Amer. Math. Soc. 139 (1969), 163-166. MR 39 #757. MR 0239400 (39:757)
  • [4] M. M. Day, Normed linear spaces, 2nd rev. ed., Ergebnisse der Mathematik, Heft 21, Springer-Verlag, Berlin, 1958. MR 20 #1187. MR 0094675 (20:1187)
  • [5] R. J. Silverman and Ti Yen, The Hahn-Banach theorem and the least upper bound property, Trans. Amer. Math. Soc. 90 (1959), 523-526. MR 21 #1511. MR 0102725 (21:1511)
  • [6] P. C. Hammer, Maximal convex sets, Duke Math. J. 22 (1955), 103-106. MR 16, 612. MR 0066662 (16:612e)
  • [7] V. L. Klee, Jr., The structure of semispaces, Math. Scand. 4 (1956), 54-64. MR 18, 330. MR 0080943 (18:330f)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A40

Retrieve articles in all journals with MSC: 46A40


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

American Mathematical Society