Ordered vector space

A point x in R2 and the set of all y such that xy (in red). The order here is xy if and only if x1y1 and x2y2.

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



Given a vector space V over the real numbers R and a partial order ≤ on the set V, the pair (V, ≤) is called an ordered vector space if for all x,y,z in V and 0 ≤ λ in R the following two axioms are satisfied

  1. xy implies x + zy + z
  2. yx implies λ y ≤ λ x.


The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping f(x) = − x is an isomorphism to the dual order structure.

If ≤ is only a preorder, (V, ≤) is called a preordered vector space.

Ordered vector spaces are ordered groups.

Positive cone

Given an ordered vector space V, the subset V+ of all elements x in V satisfying x≥0 is a convex cone, called the positive cone of V. V+ has the property that V+∩(−V+)={0}, so V+ is a proper cone. That it is convex can be seen by combining the above two axioms with the transitivity property of the (pre)order.

If V is a real vector space and C is a proper convex cone in V, there exists exactly one partial order on V that makes V into an ordered vector space such V+=C. This partial order is given by

xy if and only if yx is in C.

Therefore, there exists a one-to-one correspondence between the partial orders on a vector space V that are compatible with the vector space structure and the proper convex cones of V.


  • The real numbers with the usual order is an ordered vector space.
  • R2 is an ordered vector space with the ≤ 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 bd). 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 ac and bd (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.
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.
  • Rn is an ordered vector space with the ≤ relation defined similarly. For example, for the second order mentioned above:
    • xy if and only if xiyi for i = 1, … , n.
  • A Riesz space is an ordered vector space where the order gives rise to a lattice.
  • The space of continuous function on [0,1] where fg iff f(x) ≤ g(x) for all x in [0,1]


  • An interval in a partially ordered vector space is a convex set. If [a,b] = { x : axb }, from axioms 1 and 2 above it follows that x,y in [a,b] and λ in (0,1) implies λx+(1-λ)y in [a,b].


  • Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
  • Schaefer, Helmut H; Wolff, M.P. (1999). Topological vector spaces, 2nd ed. New York: Springer. pp. 204–205. ISBN 0387987266. 
  • Aliprantis, Charalambos D; Burkinshaw, Owen (2003). Locally solid Riesz spaces with applications to economics (Second ed.). Providence, R. I.: American Mathematical Society. ISBN 0821834088. 

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… …   Wikipedia

  • Orientation (vector space) — See also: orientation (geometry) The left handed orientation is shown on the left, and the right handed on the right. In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or… …   Wikipedia

  • Topological vector space — In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a… …   Wikipedia

  • Locally convex topological vector space — In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) which generalize normed spaces. They can be defined as topological vector… …   Wikipedia

  • Ordered group — In abstract algebra, an ordered group is a group (G,+) equipped with a partial order ≤ which is translation invariant ; in other words, ≤ has the property that, for all a , b , and g in G , if a ≤ b then a+g ≤ b+g and g+a ≤ g+b . Note that… …   Wikipedia

  • Ordered field — In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by… …   Wikipedia

  • Vector — may refer to: In mathematics * Euclidean vector, a geometric entity endowed with both length and direction, an element of a Euclidean vector space * Coordinate vector, in linear algebra, an explicit representation of an element of any abstract… …   Wikipedia

  • Vector notation — This page is an overview of the common notations used when working with vectors, which may be spatial or more abstract members of vector spaces.The common typographic convention for representing a vector is upright boldface type, as in v for a… …   Wikipedia

  • Space Shuttle Solid Rocket Booster — The Space Shuttle Solid Rocket Boosters (SRBs) are the pair of large solid rockets used by the Space Shuttle during the first two minutes of powered flight. They are located on either side of the orange external propellant tank. Each SRB produces …   Wikipedia

  • Vector Motors — Infobox Company company name = Vector Motors Corporation company company type = Private foundation = 1971 location = Wilmington, California key people = Gerald Wiegert Chairman CEO industry = Automotive products = The Vector Vector W2 Vector W8… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.