Journal of Convex Analysis 24 (2017), No. 1, 169--183
Copyright Heldermann Verlag 2017
Archimedean Cones in Vector Spaces
Eduard Yu. Emelyanov
Dept. of Mathematics, Middle East Technical University, 06800 Ankara, Turkey
eduard@metu.edu.tr
[Abstract-pdf]
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\ $(V,V_+)$ is Archimedean if and only if $$ \inf\limits_{\tau\in\{\tau\},\ y\in L}(x_\tau -y)\ =0 $$ for any bounded below decreasing net $\{x_{\tau}\}_{\tau}$ in $V$, where $L$ is the collection of all lower bounds of $\{x_\tau\}_{\tau}$, and give characterization of the almost Archimedean property of $V_+$ in terms of existence of a linear extension of an additive mapping $T:U_+\to V_+$.
Keywords: Ordered vector space, Pre-ordered vector space, Archimedean, Archimedean element, almost Archimedean, Archimedeanization, Linear extension.
MSC: 46A40
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