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

A point in and the set of all such that (in red). The order here is if and only if and

Definition

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Given a vector space   over the real numbers   and a preorder   on the set   the pair   is called a preordered vector space and we say that the preorder   is compatible with the vector space structure of   and call   a vector preorder on   if for all   and   with   the following two axioms are satisfied

  1.   implies  
  2.   implies  

If   is a partial order compatible with the vector space structure of   then   is called an ordered vector space and   is called a vector partial order on   The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping   is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation. Note that   if and only if  

Positive cones and their equivalence to orderings

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A subset   of a vector space   is called a cone if for all real     that is, for all   we have  . A cone is called pointed if it contains the origin. A cone   is convex if and only if   The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones). A cone   in a vector space   is said to be generating if  [1]

Given a preordered vector space   the subset   of all elements   in   satisfying   is a pointed convex cone (that is, a convex cone containing  ) called the positive cone of   and denoted by   The elements of the positive cone are called positive. If   and   are elements of a preordered vector space   then   if and only if   The positive cone is generating if and only if   is a directed set under   Given any pointed convex cone   one may define a preorder   on   that is compatible with the vector space structure of   by declaring for all   that   if and only if   the positive cone of this resulting preordered vector space is   There is thus a one-to-one correspondence between pointed convex cones and vector preorders on  [1] If   is preordered then we may form an equivalence relation on   by defining   is equivalent to   if and only if   and   if   is the equivalence class containing the origin then   is a vector subspace of   and   is an ordered vector space under the relation:   if and only there exist   and   such that  [1]

A subset of   of a vector space   is called a proper cone if it is a convex cone satisfying   Explicitly,   is a proper cone if (1)   (2)   for all   and (3)  [2] The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone   in a real vector space induces an order on the vector space by defining   if and only if   and furthermore, the positive cone of this ordered vector space will be   Therefore, there exists a one-to-one correspondence between the proper convex cones of   and the vector partial orders on  

By a total vector ordering on   we mean a total order on   that is compatible with the vector space structure of   The family of total vector orderings on a vector space   is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion.[1] A total vector ordering cannot be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1.[1]

If   and   are two orderings of a vector space with positive cones   and   respectively, then we say that   is finer than   if  [2]

Examples

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The real numbers with the usual ordering form a totally ordered vector space. For all integers   the Euclidean space   considered as a vector space over the reals with the lexicographic ordering forms a preordered vector space whose order is Archimedean if and only if  .[3]

Pointwise order

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If   is any set and if   is a vector space (over the reals) of real-valued functions on   then the pointwise order on   is given by, for all     if and only if   for all  [3]

Spaces that are typically assigned this order include:

  • the space   of bounded real-valued maps on  
  • the space   of real-valued sequences that converge to  
  • the space   of continuous real-valued functions on a topological space  
  • for any non-negative integer   the Euclidean space   when considered as the space   where   is given the discrete topology.

The space   of all measurable almost-everywhere bounded real-valued maps on   where the preorder is defined for all   by   if and only if   almost everywhere.[3]

Intervals and the order bound dual

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An order interval in a preordered vector space is set of the form   From axioms 1 and 2 above it follows that   and   implies   belongs to   thus these order intervals are convex. A subset is said to be order bounded if it is contained in some order interval.[2] In a preordered real vector space, if for   then the interval of the form   is balanced.[2] An order unit of a preordered vector space is any element   such that the set   is absorbing.[2]

The set of all linear functionals on a preordered vector space   that map every order interval into a bounded set is called the order bound dual of   and denoted by  [2] If a space is ordered then its order bound dual is a vector subspace of its algebraic dual.

A subset   of an ordered vector space   is called order complete if for every non-empty subset   such that   is order bounded in   both   and   exist and are elements of   We say that an ordered vector space   is order complete is   is an order complete subset of  [4]

Examples

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If   is a preordered vector space over the reals with order unit   then the map   is a sublinear functional.[3]

Properties

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If   is a preordered vector space then for all  

  •   and   imply  [3]
  •   if and only if  [3]
  •   and   imply  [3]
  •   if and only if   if and only if  [3]
  •   exists if and only if   exists, in which case  [3]
  •   exists if and only if   exists, in which case for all  [3]
    •   and
    •  
    •  
  •   is a vector lattice if and only if   exists for all  [3]

Spaces of linear maps

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A cone   is said to be generating if   is equal to the whole vector space.[2] If   and   are two non-trivial ordered vector spaces with respective positive cones   and   then   is generating in   if and only if the set   is a proper cone in   which is the space of all linear maps from   into   In this case, the ordering defined by   is called the canonical ordering of  [2] More generally, if   is any vector subspace of   such that   is a proper cone, the ordering defined by   is called the canonical ordering of  [2]

Positive functionals and the order dual

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A linear function   on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:

  1.   implies  
  2. if   then  [3]

The set of all positive linear forms on a vector space with positive cone   called the dual cone and denoted by   is a cone equal to the polar of   The preorder induced by the dual cone on the space of linear functionals on   is called the dual preorder.[3]

The order dual of an ordered vector space   is the set, denoted by   defined by   Although   there do exist ordered vector spaces for which set equality does not hold.[2]

Special types of ordered vector spaces

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Let   be an ordered vector space. We say that an ordered vector space   is Archimedean ordered and that the order of   is Archimedean if whenever   in   is such that   is majorized (that is, there exists some   such that   for all  ) then  [2] A topological vector space (TVS) that is an ordered vector space is necessarily Archimedean if its positive cone is closed.[2]

We say that a preordered vector space   is regularly ordered and that its order is regular if it is Archimedean ordered and   distinguishes points in  [2] This property guarantees that there are sufficiently many positive linear forms to be able to successfully use the tools of duality to study ordered vector spaces.[2]

An ordered vector space is called a vector lattice if for all elements   and   the supremum   and infimum   exist.[2]

Subspaces, quotients, and products

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Throughout let   be a preordered vector space with positive cone  

Subspaces

If   is a vector subspace of   then the canonical ordering on   induced by  's positive cone   is the partial order induced by the pointed convex cone   where this cone is proper if   is proper.[2]

Quotient space

Let   be a vector subspace of an ordered vector space     be the canonical projection, and let   Then   is a cone in   that induces a canonical preordering on the quotient space   If   is a proper cone in  then   makes   into an ordered vector space.[2] If   is  -saturated then   defines the canonical order of  [1] Note that   provides an example of an ordered vector space where   is not a proper cone.

If   is also a topological vector space (TVS) and if for each neighborhood   of the origin in   there exists a neighborhood   of the origin such that   then   is a normal cone for the quotient topology.[1]

If   is a topological vector lattice and   is a closed solid sublattice of   then   is also a topological vector lattice.[1]

Product

If   is any set then the space   of all functions from   into   is canonically ordered by the proper cone  [2]

Suppose that   is a family of preordered vector spaces and that the positive cone of   is   Then   is a pointed convex cone in   which determines a canonical ordering on     is a proper cone if all   are proper cones.[2]

Algebraic direct sum

The algebraic direct sum   of   is a vector subspace of   that is given the canonical subspace ordering inherited from  [2] If   are ordered vector subspaces of an ordered vector space   then   is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of   onto   (with the canonical product order) is an order isomorphism.[2]

Examples

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  • The real numbers with the usual order is an ordered vector space.
  •   is an ordered vector space with the   relation defined in any of the following ways (in order of increasing strength, that is, decreasing sets of pairs):
    • Lexicographical order:   if and only if   or   This is a total order. The positive cone is given by   or   that is, in polar coordinates, the set of points with the angular coordinate satisfying   together with the origin.
    •   if and only if   and   (the product order of two copies of   with  ). This is a partial order. The positive cone is given by   and   that is, in polar coordinates   together with the origin.
    •   if and only if   or   (the reflexive closure of the direct product of two copies of   with "<"). This is also a partial order. The positive cone is given by   or   that is, in polar coordinates,   together with the origin.
Only the second order is, as a subset of   closed; see partial orders in topological spaces.
For the third order the two-dimensional "intervals"   are open sets which generate the topology.
  •   is an ordered vector space with the   relation defined similarly. For example, for the second order mentioned above:
    •   if and only if   for  
  • A Riesz space is an ordered vector space where the order gives rise to a lattice.
  • The space of continuous functions on   where   if and only if   for all   in  

See also

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References

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  1. ^ a b c d e f g h Schaefer & Wolff 1999, pp. 250–257.
  2. ^ a b c d e f g h i j k l m n o p q r s t u Schaefer & Wolff 1999, pp. 205–209.
  3. ^ a b c d e f g h i j k l m Narici & Beckenstein 2011, pp. 139–153.
  4. ^ Schaefer & Wolff 1999, pp. 204–214.

Bibliography

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  • Aliprantis, Charalambos D; Burkinshaw, Owen (2003). Locally solid Riesz spaces with applications to economics (Second ed.). Providence, R. I.: American Mathematical Society. ISBN 0-8218-3408-8.
  • Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
  • 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.
  • Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.