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 H. H. Schaefer, "most of the ordered vector spaces occurring in analysis do not have order units." ^[2]

Definition

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 sublinear functional. ^[4]

Order unit norm

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

1. Fuchssteiner, Benno; Lusky, Wolfgang (1981). Convex Cones. Elsevier. ISBN   9780444862907.
2. Schaefer & Wolff 1999, pp. 230–234.
3. Charalambos D. Aliprantis; Rabee Tourky (2007). Cones and Duality. American Mathematical Society. ISBN   9780821841464.
4. Narici & Beckenstein 2011, pp. 139–153.

Bibliography

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN   978-1584888666. OCLC   144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN   978-1-4612-7155-0. OCLC   840278135.