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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


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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