---
tag: CFZ44H5TDZ
kind: concept
name: Archimedean ordered vector space
aliases: [archimedean-ordered]
sources:
  - wikipedia: en:63989376@r1348213680
    locator: Archimedean ordered vector space
    note: adopted
    mention: wikipedia://en/63989376
    sections: [Summary]
same_as:
  wikidata: Q96372310
  wikipedia: Archimedean ordered vector space
inventory: [HQ4SM9GNRC]
---

## Summary

In mathematics, specifically in order theory, a [[binary relation|05AVARPXA2]] $\,\leq\,$ on a [[vector space|000AYPA9AJ]] $X$ over the [[real|08J2M1JYAG]] or [[complex numbers|09FTAET5Z4]] is called **Archimedean** if for all $x \in X,$ whenever there exists some $y \in X$ such that $n x \leq y$ for all positive integers $n,$ then necessarily $x \leq 0.$ An **Archimedean (pre)ordered vector space** is a (pre)ordered vector space whose order is Archimedean. A preordered vector space $X$ is called **almost Archimedean** if for all $x \in X,$ whenever there exists a $y \in X$ such that $-n^{-1} y \leq x \leq n^{-1} y$ for all positive integers $n,$ then $x = 0.$