User:MikeRumex/sandbox/ordered vector space



In mathematics an ordered vector space is a vector space equipped with a preorder or partial order which is compatible with the vector space operations.

Definition
Given a vector space V over the real numbers R and a preorder &le; on the set V, the pair (V, &le;) is called an preordered vector space if for all x,y,z in V and 0 &le; &lambda;  in R the following two axioms are satisfied


 * 1)  x &le; y implies x + z &le; y + z
 * 2) y &le; x implies &lambda; y &le; &lambda; x.

If, in addition, &le; is a partial order, then (V, &le;) is called an partially ordered vector space.

Cones and preorders
If V is a vector space over the real numbers, a subset $$C\subseteq V$$ is called a cone if


 * 1) $$C+C\subseteq C$$
 * 2) $$\lambda C\subseteq C$$ for all $$\lambda \geq 0 $$

The set $$C\subseteq V$$ is called a proper cone if $$C\cap C = \{0\}$$.

If (V, &le;) is a preordered vector space the set $$V_+:=\{v\in V:v\geq 0\}$$ is a cone, and if &le; is a partial order, $$V_+$$ is a proper cone. Conversely, if $$C\subseteq V$$ is a cone and defining $$x\leq_C y$$ to mean $$y-x \in C$$, then $$(V,\leq _C)$$ is a preordered vector space, and a partially ordered vector space if $$C$$ is a proper cone. In this way there is a bijection between all cones (proper cones) and all preorders (partial orders) on V that are translation invariant and positively homogeneous.

Examples

 * The real numbers with the usual order is an ordered vector space.
 * R2 is an ordered vector space with the &le; relation defined in any of the following ways (in order of increasing strength, i.e., decreasing sets of pairs):
 * Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order. The positive cone is given by x > 0 or (x = 0 and y ≥ 0), i.e., in polar coordinates, the set of points with the angular coordinate satisfying -π/2 < θ ≤ π/2, together with the origin.
 * (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order of two copies of R with "≤"). This is a partial order. The positive cone is given by x ≥ 0 and y ≥ 0, i.e., in polar coordinates 0 ≤ θ ≤ π/2, together with the origin.
 * (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of two copies of R with "<"). This is also a partial order. The positive cone is given by (x > 0 and y > 0) or (x = y = 0), i.e., in polar coordinates, 0 < θ < π/2, together with the origin.
 * (a,b) ≤ (c,d) if and only if a ≤ c, is a preorder.
 * Only the second order is, as a subset of R4, closed, see partial orders in topological spaces.
 * For the third order the two-dimensional "intervals" p < x < q are open sets which generate the topology.


 * A Riesz space is a partially ordered vector space where the order gives rise to a lattice, i.e., for every pair of elements there exists a supremum.
 * The space of continuous functions on [0,1] where f ≤ g iff f(x) ≤ g(x) for all x in [0,1] is a Riesz space. When this space is also endowed with the uniform norm this space becomes a Banach lattice.
 * Rn becomes a partially ordered vector space when endowed with the standard order, i.e., x &le; y if and only if xi &le; yi for all i = 1, &hellip;, n.
 * R3 with partial order $$\leq_C$$ defined from the Lorentz cone (aka ice-cream cone) $$C:=\{x:x_1\geq \sqrt{x_2^2+x_3^2}\}$$, i.e., $$x\leq_C y$$ means $$y-x \in C$$, is a partially ordered vector space. This is one of the simplest partially ordered vector spaces that is not a vector lattice. Suprema of arbitrary pairs of elements do not necessarily exist, e.g., the pair of vectors (0,0,0) and (0,0,1) have no supremum with respect to the partial order $$\leq_C$$.