en>Dude1818: /* See also */

2013-10-16T21:26:16Z

See also

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	~~Hi there, I am Felicidad Oquendo~~. ~~After becoming out~~ of ~~my job for years I became~~ a ~~manufacturing~~ and ~~distribution officer but I strategy~~ on ~~changing it. Delaware has always been my residing location~~ and ~~will never transfer~~. The ~~factor she adores most is flower arranging~~ and ~~she~~ is ~~trying~~ to [~~http:~~//~~www~~.~~classicguides~~.~~com~~/~~automotive~~/~~extended~~-~~vehicle~~-~~warranty-plans~~/ ~~auto warranty~~] ~~make~~ ~~[http:~~//~~www~~.~~toyotacertified~~.~~com~~/~~warranty~~.~~html auto warranty~~] it a ~~occupation~~.<br><br>[~~http~~://~~www~~.~~chillister~~.~~com/blogs/post/27388 extended car warranty~~] [~~http~~:/~~/Extended~~-~~car~~-~~warranty~~-~~services-review~~.~~toptenreviews~~.~~com warranty Visit] my site car [http~~:~~//Www~~.~~gmcertified~~.~~com/used~~-~~car~~-~~warranty warranty~~] ([~~http~~:~~//www.Thenewhireguide.com/ActivityFeed/MyProfile/tabid/61/UserId/2133/Default.aspx click over here~~])		[[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]]

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