Ordered vector space

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 x in R² and the set of all y such that x ≤ y (in red). The order here is x ≤ y if and only if x₁ ≤ y₁ and x₂ ≤ y₂.

Definition

Given a vector space X over the real numbers R and a preorder ≤ on the set X, the pair (X, ≤) is called a preordered vector space and we say that the preorder ≤ is compatible with the vector space structure of X and call ≤ a vector preorder on X if for all x, y, z in X and 0 ≤ λ in ${\displaystyle \mathbb {R} }$ the following two axioms are satisfied

1. x ≤ y implies x + z ≤ y + z
2. y ≤ x implies λy ≤ λx.

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

Positive cones and their equivalence to orderings

A subset C of a vector space X is called a cone if for all real r > 0, rC ⊆ C. A cone is called pointed if it contains the origin. A cone C is convex if and only if C + C ⊆ C. 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 C in a vector space X is said to be generating if X = C − C.[1] A positive cone is generating if and only if it is a directed set under ≤.

Given a preordered vector space X, the subset X⁺ of all elements x in (X, ≤) satisfying x ≥ 0 is a pointed convex cone with vertex 0 (i.e. it contains 0) called the positive cone of X and denoted by ${\displaystyle \operatorname {PosCone} X}$. The elements of the positive cone are called positive. If x and y are elements of a preordered vector space (X, ≤), then x ≤ y if and only if y − x ∈ X⁺. Given any pointed convex cone C with vertex 0, one may define a preorder ≤ on X that is compatible with the vector space structure of X by declaring for all x and y in X, that x ≤ y if and only if y − x ∈ C; the positive cone of this resulting preordered vector space is C. There is thus a one-to-one correspondence between pointed convex cones with vertex 0 and vector preorders on X.[1] If X is preordered then we may form an equivalence relation on X by defining x is equivalent to y if and only if x ≤ y and y ≤ x; if N is the equivalence class containing the origin then N is a vector subspace of X and X/N is an ordered vector space under the relation: A ≤ B if and only there exist a in A and b in B such that a ≤ b.[1]

A subset of C of a vector space X is called a proper cone if it is a convex cone of vertex 0 satisfying C ∩ (−C) = {0}. Explicitly, C is a proper cone if (1) C + C ⊆ C, (2) rC ⊆ C for all r > 0, and (3) C ∩ (−C) = {0}.[2] The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone C in a real vector space induces an order on the vector space by defining x ≤ y if and only if y − x ∈ C, and furthermore, the positive cone of this ordered vector space will be C. Therefore, there exists a one-to-one correspondence between the proper convex cones of X and the vector partial orders on X.

By a total vector ordering on X we mean a total order on X that is compatible with the vector space structure of X. The family of total vector orderings on a vector space X 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 R and S are two orderings of a vector space with positive cones P and Q, respectively, then we say that R is finer than S if P ⊆ Q.[2]

Examples

The real numbers with the usual ordering form a totally ordered vector space. For all integers n ≥ 0, the Euclidean space ℝⁿ 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 n = 0 or 1.[3]

Pointwise order

If S is any set and if X is a vector space (over the reals) of real-valued functions on S, then the pointwise order on X is given by, for all f, g ∈ X, f ≤ g if and only if f(s) ≤ g(s) for all s in S.[3]

Spaces that are typically assigned this order include:

• the space 𝓁_∞(S, ℝ) of bounded real-valued maps on S.
• the space c₀(ℝ) of real-valued sequences that converge to 0.
• the space C(S, ℝ) of continuous real-valued functions on a topological space S.
• for any non-negative integer n, the Euclidean space ℝⁿ when considered as the space C({1, ..., n}, ℝ) where S = {1, ..., n} is given the discrete topology.

The space ${\displaystyle {\mathcal {L}}^{\infty }\left(\mathbb {R} ,\mathbb {R} \right)}$ of all measurable almost-everywhere bounded real-valued maps on ℝ, where the preorder is defined for all f, g ∈ ${\displaystyle {\mathcal {L}}^{\infty }\left(\mathbb {R} ,\mathbb {R} \right)}$ by f ≤ g if and only if f(s) ≤ g(s) almost everywhere.[3]

Intervals and the order bound dual

An order interval in a preordered vector space is set of the form

[a, b] = {x : a ≤ x ≤ b},

[a, b[ = {x : a ≤ x < b},

]a, b] = {x : a < x ≤ b}, or

]a, b[ = {x : a < x < b}.

From axioms 1 and 2 above it follows that x, y ∈ [a, b] and 0 < λ < 1 implies λx + (1 − λ)y in [a, b]; 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 x ≥ 0 then the interval of the form [−x, x] is balanced.[2] An order unit of a preordered vector space is any element x such that the set [−x, x] is absorbing.[2]

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

A subset A of an ordered vector space X is called order complete if for every non-empty subset B ⊆ A such that B is order bounded in A, both ${\displaystyle \sup B}$ and ${\displaystyle \inf B}$ exist and are elements of A. We say that an ordered vector space X is order complete is X is an order complete subset of X.[4]

Examples

If (X, ≤) is a preordered vector space over the reals with order unit u, then the map ${\displaystyle p(x):=\inf \left\{t\in \mathbb {R} :x\leq tu\right\}}$ is a sublinear functional.[3]

Properties

If X is a preordered vector space then for all x, y ∈ X,

• x ≥ 0 and y ≥ 0 imply x + y ≥ 0.[3]
• x ≤ y if and only if −y ≤ −x.[3]
• x ≤ y and r < 0 imply rx ≥ ry.[3]
• x ≤ y if and only if y = sup{x, y} if and only if x = inf{x, y}.[3]
• sup{x, y} exists if and only if inf{−x, −y} exists, in which case inf{−x, −y} = −sup{x, y}.[3]
• sup{x, y} exists if and only if inf{x, y} exists, in which case for all z ∈ X,[3]
  • sup{x + z, y + z} = z + sup{x, y}, and
  • inf{x + z, y + z} = z + inf{x, y}
  • x + y = inf{x, y} + sup{x, y}.
• X is a vector lattice if and only if sup{0, x} exists for all x in X.[3]

Spaces of linear maps

A cone C is said to be generating if C − C is equal to the whole vector space.[2] If X and W are two non-trivial ordered vector spaces with respective positive cones P and Q, then P is generating in X if and only if the set ${\displaystyle C=\left\{u\in L(X;W):u(P)\subseteq Q\right\}}$ is a proper cone in L(X; W), which is the space of all linear maps from X into W. In this case the ordering defined by C is called the canonical ordering of L(X; W).[2] More generally, if M is any vector subspace of L(X; W) such that C ∩ M is a proper cone, the ordering defined by C ∩ M is called the canonical ordering of M.[2]

Positive functionals and the order dual

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

1. x ≥ 0 implies f(x) ≥ 0.
2. if x ≤ y then f(x) ≤ f(y).[3]

The set of all positive linear forms on a vector space with positive cone C, called the dual cone and denoted by ${\displaystyle C^{*}}$, is a cone equal to the polar of −C. The preorder induced by the dual cone on the space of linear functionals on X is called the dual preorder.[3]

The order dual of an ordered vector space X is the set, denoted by ${\displaystyle X^{+}}$, defined by ${\displaystyle X^{+}:=C^{*}-C^{*}}$. Although ${\displaystyle X^{+}\subseteq X^{b}}$, there do exist ordered vector spaces for which set equality does not hold.[2]

Special types of ordered vector spaces

Let X be an ordered vector space. We say that an ordered vector space X is Archimedean ordered and that the order of X is Archimedean if whenever x in X is such that ${\displaystyle \left\{nx:n\in \mathbb {N} \right\}}$ is majorized (i.e. there exists some y in X such that nx ≤ y for all ${\displaystyle n\in \mathbb {N} }$) then x ≤ 0.[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 X is regularly ordered and that its order is regular if it is Archimedean ordered and X⁺ distinguishes points in X.[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 x and y, the supremum sup(x, y) and infimum inf(x, y) exist.[2]

Subspaces, quotients, and products

Throughout let X be a preordered vector space with positive cone C.

Subspaces

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

Quotient space

Let M be a vector subspace of an ordered vector space X, ${\displaystyle \pi :X\to X/M}$ be the canonical projection, and let ${\displaystyle {\hat {C}}:=\pi (C)}$. Then ${\displaystyle {\hat {C}}}$ is a cone in X/M that induces a canonical preordering on the quotient space X/M. If ${\displaystyle {\hat {C}}}$ is a proper cone in X/M then ${\displaystyle {\hat {C}}}$ makes X/M into an ordered vector space.[2] If M is C-saturated then ${\displaystyle {\hat {C}}}$ defines the canonical order of X/M.[1] Note that ${\displaystyle X=\mathbb {R} _{0}^{2}}$ provides an example of an ordered vector space where ${\displaystyle \pi (C)}$ is not a proper cone.

If X is also a topological vector space (TVS) and if for each neighborhood V of 0 in X there exists a neighborhood U of 0 such that [(U + N) ∩ C] ⊆ V + N then ${\displaystyle {\hat {C}}}$ is a normal cone for the quotient topology.[1]

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

Product

If S is any set then the space X^S of all functions from S into X is canonically ordered by the proper cone ${\displaystyle \left\{f\in X^{S}:f(s)\in C{\text{ for all }}s\in S\right\}}$.[2]

Suppose that ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}$ is a family of preordered vector spaces and that the positive cone of ${\displaystyle X_{\alpha }}$ is ${\displaystyle C_{\alpha }}$. Then ${\displaystyle \textstyle {C:=\prod _{\alpha }C_{\alpha }}}$ is a pointed convex cone in ${\displaystyle \textstyle {\prod _{\alpha }X_{\alpha }}}$, which determines a canonical ordering on ${\displaystyle \textstyle {\prod _{\alpha }X_{\alpha }}}$; C is a proper cone if all ${\displaystyle C_{\alpha }}$ are proper cones.[2]

Algebraic direct sum

The algebraic direct sum ${\displaystyle \textstyle {\bigoplus _{\alpha }X_{\alpha }}}$ of ${\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}}$ is a vector subspace of ${\displaystyle \textstyle {\prod _{\alpha }X_{\alpha }}}$ that is given the canonical subspace ordering inherited from ${\displaystyle \textstyle {\prod _{\alpha }X_{\alpha }}}$.[2] If X₁, ..., X_n are ordered vector subspaces of an ordered vector space X then X is the ordered direct sum of these subspaces if the canonical algebraic isomorphism of X onto ${\displaystyle \textstyle {\prod _{\alpha }X_{\alpha }}}$ (with the canonical product order) is an order isomorphism.[2]

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 x_i ≤ y_i for i = 1, ..., n.
• A Riesz space is an ordered vector space where the order gives rise to a lattice.
• The space of continuous functions on [0, 1] where f ≤ g iff f(x) ≤ g(x) for all x in [0, 1].

See also

• Ordered topological vector space
• Order topology (functional analysis)
• Partially ordered space
• Riesz space
• Topological vector lattice
• Vector lattice
• Product order

References

1. Schaefer & Wolff 1999, pp. 250-257.
2. Schaefer & Wolff 1999, pp. 205–209.
3. Narici & Beckenstein 2011, pp. 139-153.
4. Schaefer & Wolff 1999, pp. 204-214.

Bibliography

• 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. 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.