Archimedean Cones in Vector Spaces

E. Y. Emelyanov

Journal of convex analysis, Tome 24 (2017) no. 1, pp. 169-183

Résumé

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 $T:U_+\to V_+$.

Détail

Classification : 46A40
Mots-clés : Ordered vector space, Pre-ordered vector space, Archimedean, Archimedean element, almost Archimedean, Archimedeanization, Linear extension