Order unit

An order unit is an element of an ordered vector space which can be used to bound all elements from above.^[1] In this way (as seen in the first example below) the order unit generalizes the unit element in the reals.

According to

For the ordering cone ${\displaystyle K\subseteq X}$ in the vector space ${\displaystyle X}$, the element ${\displaystyle e\in K}$ is an order unit (more precisely a ${\displaystyle K}$-order unit) if for every ${\displaystyle x\in X}$ there exists a ${\displaystyle \lambda _{x}>0}$ such that ${\displaystyle \lambda _{x}e-x\in K}$ (that is, ${\displaystyle x\leq _{K}\lambda _{x}e}$).^[3]

Equivalent definition

The order units of an ordering cone ${\displaystyle K\subseteq X}$ are those elements in the algebraic interior of ${\displaystyle K;}$ that is, given by ${\displaystyle \operatorname {core} (K).}$^[3]

Examples

Let ${\displaystyle X=\mathbb {R} }$ be the real numbers and ${\displaystyle K=\mathbb {R} _{+}=\{x\in \mathbb {R} :x\geq 0\},}$ then the unit element ${\displaystyle 1}$ is an order unit.

Let ${\displaystyle X=\mathbb {R} ^{n}}$ and ${\displaystyle K=\mathbb {R} _{+}^{n}=\left\{x_{i}\in \mathbb {R} :{\text{ for all }}i=1,\ldots ,n:x_{i}\geq 0\right\},}$ then the unit element ${\displaystyle {\vec {1}}=(1,\ldots ,1)}$ is an order unit.

Each interior point of the positive cone of an ordered topological vector space is an order unit.^[2]

Properties

Each order unit of an ordered TVS is interior to the positive cone for the order topology.^[2]

If ${\displaystyle (X,\leq )}$ is a preordered vector space over the reals with order unit ${\displaystyle u,}$ then the map ${\displaystyle p(x):=\inf\{t\in \mathbb {R} :x\leq tu\}}$ is a

Suppose ${\displaystyle (X,\leq )}$ is an ordered vector space over the reals with order unit ${\displaystyle u}$ whose order is Archimedean and let ${\displaystyle U=[-u,u].}$ Then the Minkowski functional ${\displaystyle p_{U}}$ of ${\displaystyle U,}$ defined by ${\displaystyle p_{U}(x):=\inf\{r>0:x\in r[-u,u]\},}$ is a norm called the order unit norm. It satisfies ${\displaystyle p_{U}(u)=1}$ and the closed unit ball determined by ${\displaystyle p_{U}}$ is equal to ${\displaystyle [-u,u];}$ that is, ${\displaystyle [-u,u]=\left\{x\in X:p_{U}(x)\leq 1\right\}.}$^[4]

References