A new proof of the equivalence of the Hahn-Banach extension and the least upper bound properties
HTML articles powered by AMS MathViewer
- by A. D. Ioffe PDF
- Proc. Amer. Math. Soc. 82 (1981), 385-389 Request permission
Abstract:
The paper contains a new proof of the fact that the Hahn-Banach majorized extension theorem for linear operators is valid iff the range ordered space is conditionally complete. The proof is based on quite different principles than the original proof of Bonnice, Silverman and To. The key element is a reformulation of the extension problem in terms of linear selections of special convex-valued mappings called fans.References
- William E. Bonnice and Robert J. Silverman, The Hahn-Banach theorem for finite dimensional spaces, Trans. Amer. Math. Soc. 121 (1966), 210–222. MR 185412, DOI 10.1090/S0002-9947-1966-0185412-0
- 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 215050, DOI 10.1090/S0002-9939-1967-0215050-9
- Mahlon M. Day, Normed linear spaces, Reihe: Reelle Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0094675
- Alexandre D. Ioffe, Différentielles généralisées d’applications localement lipschitziennes d’un espace de Banach dans un autre, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 13, A637–A640 (French, with English summary). MR 556447
- Alexander D. Ioffe, On foundations of convex analysis, Third International Conference on Collective Phenomena (Moscow, 1978), Ann. New York Acad. Sci., vol. 337, New York Acad. Sci., New York, 1980, pp. 103–117. MR 624285 L. V. Kantorovič, On semi-ordered linear spaces and their applications to the theory of linear operators, Dokl. Akad. Nauk SSSR 4 (1935), 11-14.
- B. Rodríguez-Salinas and L. Bou, A Hahn-Banach theorem for arbitrary vector spaces, Boll. Un. Mat. Ital. (4) 10 (1974), 390–393 (English, with Italian summary). MR 0365079
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- 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 102725, DOI 10.1090/S0002-9947-1959-0102725-5
- Ting-On To, The equivalence of the least upper bound property and the Hahn-Banach extension property in ordered linear spaces, Proc. Amer. Math. Soc. 30 (1971), 287–295. MR 417746, DOI 10.1090/S0002-9939-1971-0417746-3
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 385-389
- MSC: Primary 46A22; Secondary 46A40
- DOI: https://doi.org/10.1090/S0002-9939-1981-0612725-1
- MathSciNet review: 612725