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Some order and topological properties of locally solid linear topological Riesz spaces


Author: C. D. Aliprantis
Journal: Proc. Amer. Math. Soc. 40 (1973), 443-447
MSC: Primary 46A40
DOI: https://doi.org/10.1090/S0002-9939-1973-0320692-9
MathSciNet review: 0320692
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Abstract: A theorem of Luxemburg and Zaanen on normed Riesz spaces (Theorem 2.4 below) and one of Nakano (Theorem 2.3 below) have been extended by the author in [1] to metrizable locally solid linear topological Riesz spaces. This note gives an example which shows they cannot be further extended to nonmetrizable Hausdorff locally solid linear topological Riesz spaces.


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

  • [1] C. D. Aliprantis, On order and topological properties of Riesz spaces, Thesis, California Institute of Technology, 1973.
  • [2] J. Hornáth, Topological vector spaces and distributions. I, Addison-Wesley, London, 1966.
  • [3] W. A. J. Luxemburg, Notes on Banach function spaces. XlVb, Nederl. Akad. Wetensch. Proc. Ser. A68 = Indag. Math. 27 (1965), 240-248. MR 32 #6202b. MR 0188767 (32:6202b)
  • [4] W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces. X, XI, Nederl. Akad. Wetensch. Proc. Ser. A67 = Indag. Math. 26 (1964), 493-505. MR 30 #3381b. MR 0173168 (30:3381b)
  • [5] -, Riesz spaces. I, North-Holland, Amsterdam, 1971.
  • [6] H. Nakano, Semi-ordered linear spaces, Japan Society for the Promotion of Science, Tokyo, 1955. MR 17, 387. MR 0073124 (17:387b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0320692-9
Keywords: Locally solid Riesz spaces, order and topological continuity
Article copyright: © Copyright 1973 American Mathematical Society

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