Abstract

In the case of an ordered vector space (briefly, OVS) with an order unit, the Archimedeanization method was recently developed by V. I. Paulsen and M. Tomforde [Vector spaces with an order unit, Indiana Univ. Math. J. 58(3) (2009) 1319–1359]. We present a general version of the Archimedeanization which covers arbitrary OVS. Also we show that an OVS (V,V+)(V,V_+) is Archimedean if and only if inf⁡τ∈{τ}, y∈L(xτ−y) =0\inf\limits_{\tau\in\{\tau\},\ y\in L}(x_\tau -y)\ =0 for any bounded below decreasing net {xτ}τ\{x_{\tau}\}_{\tau} in VV, where LL is the collection of all lower bounds of {xτ}τ\{x_\tau\}_{\tau}, and give characterization of the almost Archimedean property of V+V_+ in terms of existence of a linear extension of an additive mapping T:U+→V+T:U_+\to V_+.

Author information

Contact details are reproduced from the original publication and may be historical.

Eduard Yu. Emelyanov

Dept. of Mathematics, Middle East Technical University, 06800 Ankara, Turkey
and: Sobolev Institute of Mathematics of the Russian Academy of Sciences, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia

eduard@metu.edu.tr

Keywords

  • Ordered vector space
  • Pre-ordered vector space
  • Archimedean
  • Archimedean element
  • almost Archimedean
  • Archimedeanization
  • Linear extension

Mathematics Subject Classification

46A40

References

  1. C. D. Aliprantis, R. Tourky: Cones and Duality, Graduate Studies in Mathematics 84, American Mathematical Society, Providence (2007).https://doi.org/10.1090/gsm/084Google Scholar
  2. E. Yu. Emel’yanov: Infinitesimals in ordered vector spaces, Vladikavkaz. Mat. Zh. 15(1) (2013) 19–24.https://doi.org/10.23671/VNC.2013.1.10504Google Scholar
  3. W. A. J. Luxemburg, A. C. Zaanen: Riesz Spaces I, North-Holland, Amsterdam (1971).Google Scholar
  4. V. I. Paulsen, M. Tomforde: Vector spaces with an order unit, Indiana Univ. Math. J. 58(3) (2009) 1319–1359.https://doi.org/10.1512/iumj.2009.58.3518Google Scholar
  5. A. I. Veksler: Archimedean principle in homomorphic images of l-groups and of vector lattices, Izv. Vysš. Učebn. Zaved. Mat. 54(4) (1966) 33–38.Google Scholar
  6. A. C. Zaanen: Riesz Spaces II, North-Holland, Amsterdam (1983).Google Scholar

Suggested citation

E. Y. Emelyanov. “Archimedean Cones in Vector Spaces.” Journal of Convex Analysis 24 (2017), No. 1, 169–183. https://doi.org/10.68381/jca24013

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.