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[[File:Ordered space illustration.svg|right|thumb|A point ''x'' in '''R'''<sup>2</sup> and the set of all ''y'' such that ''x''≤''y'' (in red). The order here is ''x''≤''y'' if and only if ''x''<sub>''1''</sub> ≤ ''y''<sub>''1''</sub> and ''x''<sub>''2''</sub> ≤ ''y''<sub>''2''</sub>.]] | |||
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. | |||
==Definition== | |||
Given a vector space ''V'' over the [[real number]]s '''R''' and a [[partial order]] ≤ on the [[set (mathematics)|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 | |||
# ''x'' ≤ ''y'' implies ''x'' + ''z'' ≤ ''y'' + ''z'' | |||
# ''y'' ≤ ''x'' implies λ ''y'' ≤ λ ''x''. | |||
==Notes== | |||
The two axioms imply that [[Translation (geometry)|translation]]s and [[positive homothety|positive homotheties]] are automorphisms of the order structure and the mapping ''f''(''x'') = − ''x'' is an isomorphism to the [[Duality (order theory)|dual order structure]]. | |||
If ≤ is only a [[preorder]], (''V'', ≤) is called a '''preordered vector space'''. | |||
Ordered vector spaces are [[ordered group]]s under their addition operation. | |||
==Positive cone== | |||
Given an ordered vector space ''V'', the subset ''V''<sup>+</sup> of all elements ''x'' in ''V'' satisfying ''x''≥0 is a [[convex cone]], called the '''positive cone''' of ''V''. Since the [[partial order]] ≥ is [[Antisymmetric_relation|antisymmetric]], one can show, that ''V''<sup>+</sup>∩(−''V''<sup>+</sup>)={0}, hence ''V''<sup>+</sup> 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 <math>V</math> that makes ''V'' into an ordered vector space such ''V''<sup>+</sup>=''C''. This partial order is given by | |||
: ''x'' ≤ ''y'' if and only if ''y''−''x'' 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''. | |||
== Examples == | |||
* The [[real number]]s with the usual order is an ordered vector space. | |||
* '''R'''<sup>2</sup> 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 ''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 coordinate system|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#Direct_product_of_binary_relations|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 '''R'''<sup>4</sup>, closed, see [[Partially_ordered_set#Partial_orders_in_topological_spaces|partial orders in topological spaces]]. | |||
:For the third order the two-dimensional "[[Partially_ordered_set#Interval|intervals]]" ''p'' < ''x'' < ''q'' are open sets which generate the topology. | |||
* '''R'''<sup>''n''</sup> is an ordered vector space with the ≤ relation defined similarly. For example, for the second order mentioned above: | |||
**''x'' ≤ ''y'' if and only if ''x''<sub>''i''</sub> ≤ ''y''<sub>''i''</sub> for ''i'' = 1, … , ''n''. | |||
* A [[Riesz space]] is an ordered vector space where the order gives rise to a [[Lattice order|lattice]]. | |||
* The space of continuous function on [0,1] where ''f'' ≤ ''g'' iff f(x) ≤ g(x) for all x in [0,1] | |||
== Remarks == | |||
* An interval in a partially ordered vector space is a [[convex set]]. If [''a'',''b''] = { ''x'' : ''a'' ≤ ''x'' ≤ ''b'' }, from axioms 1 and 2 above it follows that ''x'',''y'' in [''a'',''b''] and λ in (0,1) implies λ''x''+(1-λ)''y'' in [''a'',''b'']. | |||
== See also == | |||
* [[Partially ordered space]] | |||
* [[Riesz space]] | |||
== References == | |||
* [[Bourbaki, Nicolas]]; <cite>Elements of Mathematics: Topological Vector Spaces</cite>; ISBN 0-387-13627-4. | |||
*{{cite book | |||
| authorlink = Helmut Schaefer | |||
| last = Schaefer | |||
| first = Helmut H | |||
| coauthors = Wolff, M.P. | |||
| title = Topological vector spaces, 2nd ed | |||
| publisher = New York: Springer | |||
| date = 1999 | |||
| pages = 204–205 | |||
| isbn = 0-387-98726-6 | |||
}} | |||
*{{cite book | |||
| last = Aliprantis | |||
| first = Charalambos D | |||
| authorlink = Charalambos D. Aliprantis | |||
| coauthors = Burkinshaw, Owen | |||
| title = Locally solid Riesz spaces with applications to economics | |||
| edition = Second | |||
| publisher = Providence, R. I.: American Mathematical Society | |||
| date = 2003 | |||
| pages = | |||
| isbn = 0-8218-3408-8 | |||
}} | |||
[[Category:Functional analysis]] | |||
[[Category:Ordered groups]] | |||
Latest revision as of 22:26, 16 October 2013
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.
Definition
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
- x ≤ y implies x + z ≤ y + z
- y ≤ x implies λ y ≤ λ x.
Notes
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 under their addition operation.
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. Since the partial order ≥ is antisymmetric, one can show, that V+∩(−V+)={0}, hence 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 that makes V into an ordered vector space such V+=C. This partial order is given by
- x ≤ y if and only if y−x 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.
Examples
- 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 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.
- 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:
- x ≤ y if and only if xi ≤ yi 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 f ≤ g iff f(x) ≤ g(x) for all x in [0,1]
Remarks
- An interval in a partially ordered vector space is a convex set. If [a,b] = { x : a ≤ x ≤ b }, 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].
See also
References
- Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
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