Ordered vector space

An ordered vector space is a mathematical structure . It is a - vector space on which there is also an order relation compatible with the vector space structure , which is usually referred to with (one reads less than or equal). This makes it possible to compare the elements of a vector space according to size . Many vector spaces examined in mathematics have a natural order structure. ${\displaystyle \mathbb {R} }$ ${\displaystyle \leq }$

definition

An ordered vector space is a pair consisting of a vector space and an order relation on , so that the following applies: ${\displaystyle (V,\leq )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle \leq }$ ${\displaystyle V}$

• ${\displaystyle x\leq x}$for everyone , that is, is reflexive . ${\displaystyle x\in V}$ ${\displaystyle \leq }$
• From and follows for all , that is, is transitive . ${\displaystyle x\leq y}$ ${\displaystyle y\leq z}$ ${\displaystyle x\leq z}$ ${\displaystyle x,y,z\in V}$ ${\displaystyle \leq }$
• From it follows for all , that is, is compatible with the addition. ${\displaystyle x\leq y}$ ${\displaystyle x+z\leq y+z}$ ${\displaystyle x,y,z\in V}$ ${\displaystyle \leq }$
• It follows for all and , that is, is compatible with scalar multiplication . ${\displaystyle x\leq y}$ ${\displaystyle \lambda x\leq \lambda y}$ ${\displaystyle x,y\in V}$ ${\displaystyle \lambda \in \mathbb {R} ,\lambda \geq 0}$ ${\displaystyle \leq }$

In the definition one can replace by an ordered body . In most applications, however, you are dealing with the field of real numbers. A vector space is called an ordered vector space if it is ordered as a real vector space. Many of the concepts discussed here can be generalized to ordered Abelian groups . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

Positive cone

→ Main article : Positive cone

If it is an ordered vector space, the positive cone is called . It is in fact a cone , which means that: ${\displaystyle (V,\leq )}$ ${\displaystyle V^{+}:=\{x\in V;\,x\geq 0\}}$

• For everyone and applies . ${\displaystyle x,y\in V^{+}}$ ${\displaystyle \lambda ,\mu \geq 0}$ ${\displaystyle \lambda x+\mu y\in V^{+}}$

In particular, the positive cone is convex , which gives rise to geometrical investigations.

Conversely, if a cone is given in a -vector space , then an order relation is defined, which makes it an ordered vector space, so that . An ordered vector space can therefore also be understood as a vector space with a distinct cone. Properties of order can be related to algebraic and geometrical properties of the cone; is even a topological vector space , then topological properties of the cone are added. ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle P}$ ${\displaystyle x\leq y:\Leftrightarrow y-x\in P}$ ${\displaystyle (V,\leq )}$ ${\displaystyle V^{+}=P}$ ${\displaystyle V}$

Positive operators

The structure-preserving mappings between ordered vector spaces and are the linear operators that also preserve the order structure, that is, for which it always follows. Such mappings are called positive or monotone operators. The study of positive operators is an important part of the theory of ordered vector spaces. ${\displaystyle (V,\leq )}$ ${\displaystyle (W,\preceq )}$ ${\displaystyle f\colon V\to W}$ ${\displaystyle x\leq y}$ ${\displaystyle f(x)\preceq f(y)}$

Obviously the ordered vector spaces with the positive operators as morphisms form a category .

An ordering interval is a set of form . A linear operator between ordered vector spaces is called order restricted if it maps order intervals into order intervals. Differences of positive operators are obviously order restricted. ${\displaystyle [x,y]:=\{z\in V;x\leq z\leq y\}}$

Dual order

If is an ordered vector space, then is a cone that makes the dual space an ordered vector space; this is the so-called dual order on . Is also a topological vector space, so you look instead of the algebraic topological dual space, ie the space of all continuous linear functionals on . If this space is normalized or more generally locally convex , the duality theory, which is rich for these space classes, is available. ${\displaystyle (V,\leq )}$ ${\displaystyle P:=\{f\in V\,';\,f(x)\geq 0\,\forall x\in V^{+}\}}$ ${\displaystyle V\,'}$ ${\displaystyle V\,'}$ ${\displaystyle V}$ ${\displaystyle V}$

Often one only looks at the subspace of the orderly restricted functional and speaks of the orderly restricted dual space. ${\displaystyle V^{b}\subset V\,^{'}}$

Examples

• The sequence spaces such as , or are ordered vector spaces if one explains the order component-wise, that is, if one defines for two sequences and the relation through . ${\displaystyle c}$ ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle x=(x_{n})_{n}}$ ${\displaystyle y=(y_{n})_{n}}$ ${\displaystyle x\leq y}$ ${\displaystyle x_{n}\leq y_{n}\forall n\in \mathbb {N} }$
• Function spaces like or L ^p [0,1] are ordered vector spaces if the order is explained point by point, that is, if one defines for two functions and the relation through for all from the domain or almost everywhere on the domain. ${\displaystyle C[0,1]}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f\leq g}$ ${\displaystyle f(x)\leq g(x)}$ ${\displaystyle x}$
• If one is a C * -algebra and one sets , one can show that there is a cone, which turns into an ordered vector space. The investigation of the dual space with the dual order is an important method in the theory of C * algebras. ${\displaystyle A}$ ${\displaystyle A^{+}:=\{x^{*}x;\,x\in A\}}$ ${\displaystyle A^{+}}$ ${\displaystyle A}$

More terms

Let be an ordered vector space. ${\displaystyle (V,\leq )}$

Strict order

In the definition given here it was not required that from and always should follow; the order relation would then be antisymmetric , and this would be equivalent to the cone being pointed (i.e. ). Most of the cones used in the applications are pointed. Some authors always understand a cone to be a pointed cone and call the more general term introduced above a truncated cone . Antisymmetric orders are also called strict orders. ${\displaystyle x\leq y}$ ${\displaystyle y\leq x}$ ${\displaystyle x=y}$ ${\displaystyle V^{+}}$ ${\displaystyle V^{+}\cap (-V^{+})=\{0\}}$

Directed order

The order on is called directed if there is always one with and for every two elements . The order is directed if and only if , that is, if the positive cone generates the vector space . ${\displaystyle V}$ ${\displaystyle x,y\in V}$ ${\displaystyle z\in V}$ ${\displaystyle x\leq z}$ ${\displaystyle y\leq z}$ ${\displaystyle V=V^{+}-V^{+}}$

Organizational units

An element is an atomic unit, if for every one is having . This is equivalent to the ordering interval being an absorbing quantity . ${\displaystyle e\in V}$ ${\displaystyle x\in V}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle -n\cdot e\leq x\leq n\cdot e}$ ${\displaystyle [-e,e]:=\{x\in V;-e\leq x\leq e\}}$

Obviously the constant function 1 is a unit of order in , while the sequence space has no units of order. ${\displaystyle C[0,1]}$ ${\displaystyle c_{0}}$

Archimedean order

The order in is called Archimedean if the following applies: Are and is for everyone , it follows . ${\displaystyle V}$ ${\displaystyle x,y\in V}$ ${\displaystyle nx\leq y}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x\leq 0}$

The order is almost Archimedean , if the following applies: Are and is for everyone , then it follows . ${\displaystyle x,y\in V}$ ${\displaystyle -y\leq nx\leq y}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x=0}$

The order is nowhere Archimedean , if for every one is using for all . ${\displaystyle x\in V}$ ${\displaystyle y\in V}$ ${\displaystyle -y\leq nx\leq y}$ ${\displaystyle n\in \mathbb {N} }$

Subspaces, quotients and direct products

If there is an ordered vector space and a subspace, then with the restricted order there is again an ordered vector space, it is evident and the embedding is a positive operator. ${\displaystyle (V,\leq )}$ ${\displaystyle U\subset V}$ ${\displaystyle U}$ ${\displaystyle U^{+}=U\cap V^{+}}$ ${\displaystyle U\subset V}$

The quotient space obviously becomes an ordered vector space with the cone and the quotient mapping is a positive operator. ${\displaystyle V/U}$ ${\displaystyle \{x+U;\,x\in V^{+}\}}$ ${\displaystyle V\rightarrow V/U}$

Finally , if there is a family of ordered vector spaces, the direct product becomes an ordered vector space if the positive cone is explained by. An important question in the theory of ordered vector spaces is whether a given ordered vector space can be decomposed as a direct product of ordered spaces. ${\displaystyle (V_{i},\leq _{i})_{i\in I}}$ ${\displaystyle \textstyle \prod _{i\in I}V_{i}}$ ${\displaystyle \{(v_{i})_{i};\,v_{i}\in V_{i}^{+},i\in I\}}$

Riesz rooms

A strictly ordered vector space has the Riesz decomposition property if and only if the following applies: ${\displaystyle (V,\leq )}$

Is and , so there is with , and . ${\displaystyle y,x_{1},x_{2}\in V^{+}}$ ${\displaystyle 0\leq y\leq x_{1}+x_{2}}$ ${\displaystyle y_{i}\in V}$ ${\displaystyle 0\leq y_{1}\leq x_{1}}$ ${\displaystyle 0\leq y_{2}\leq x_{2}}$ ${\displaystyle y_{1}+y_{2}\,=\,y}$

If for every two elements of a strictly ordered vector space there is always a smallest element with and , which is then denoted by and the supremum and is called, then one speaks of a Riesz space or vector lattice . One can show that there is indeed a distributive lattice, whereby the other lattice operation could be defined by. It can be shown that vector lattices have the Riesz decomposition property. A vector lattice is called complete if not only two elements but every set bounded above has a supremum. ${\displaystyle x,y\in V}$ ${\displaystyle z}$ ${\displaystyle x\leq z}$ ${\displaystyle y\leq z}$ ${\displaystyle x\vee y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x\wedge y=-((-x)\vee (-y))}$

Comment on the designation: Some authors call directed and strictly ordered vector spaces with the Riesz decomposition property Riesz spaces, see for example, and therefore do not use Riesz space as a synonym for vector lattice.

In connection with the terms introduced here, there is the following important sentence by F. Riesz:

• If a directed and strictly ordered vector space with Riesz's decomposition property, then the order-restricted dual space is a complete vector lattice. ${\displaystyle (V,\leq )}$

Consider a C * algebra as an application . Then the self-adjoint part is a real vector space, which through the cone becomes a directed and strictly ordered vector space with Riesz interpolation property. The dual space , which coincides with the order-restricted dual space, is therefore a complete vector lattice, which is important for the C * theory. ${\displaystyle A}$ ${\displaystyle A_{sa}:=\{a\in A;a^{*}=a\}}$ ${\displaystyle A^{+}:=\{a^{*}a;a\in A\}}$ ${\displaystyle A_{sa}^{'}}$

Topological ordered vector spaces

If an ordered vector space also has a vector space topology, one speaks of an ordered, topological vector space and can investigate continuity properties of the order. Especially in vector lattices one can see the continuity of the mappings

• ${\displaystyle (x,y)\mapsto x\vee y}$
• ${\displaystyle x\mapsto x^{+}:=0\vee x,\quad x\mapsto x^{-}:=0\wedge x}$
• ${\displaystyle x\mapsto |x|:=x^{+}\vee x^{-}}$

to study.

The following theorem applies to ordered topological vector lattices : ${\displaystyle V}$

• The mapping is continuous if and only if has a zero neighborhood base of sets that have the following property: Is and with , then follows . ${\displaystyle V\times V\rightarrow V,(x,y)\mapsto x\vee y}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle x\in U}$ ${\displaystyle y\in V}$ ${\displaystyle |y|\leq |x|}$ ${\displaystyle y\in U}$

If there is even a normalized space with norm and a vector lattice, then the norm is called a lattice norm if it always follows. In this case one speaks of a standardized vector lattice. Then the above quoted sentence is applicable and one recognizes the continuity of the union operations. Typical examples are the examples listed above or with their natural orders and norms. ${\displaystyle V}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle |x|\leq |y|}$ ${\displaystyle \|x\|\leq \|y\|}$ ${\displaystyle c_{0},\ell ^{1}}$ ${\displaystyle C([0,1])}$

There is an extensive theory for ordered topological vector spaces, especially ordered Banach spaces, for which reference is made to the literature at this point.

See also

• Generalized inequality

Individual evidence

1. ^ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 1.1
2. ↑ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 1.1.3
3. ↑ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 1.3.1
4. ^ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 1.3
5. ↑ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 1.4
6. ^ CD Aliprantis, R. Tourky: Cones and duality , American Mathematical Society (2007), 1.14
7. ^ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 2.1
8. ^ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 2.6.1
9. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups ISBN 0125494505 , 1.4.10
10. ↑ Graham Jameson: Ordered Linear Spaces, Springer Lecture Notes, Volume 141 (1970), 4.1.5
11. ^ CD Aliprantis, R. Tourky: Cones and duality , American Mathematical Society (2007), 2.36

literature

• Graham Jameson: Ordered Linear Spaces , Springer Lecture Notes, Volume 141 (1970)
• WAJ Luxemburg and AC Zaanen: Riesz-Spaces , North-Holland Pub. Co .; New York, American Elsevier Pub. Co. (1971), ISBN 0444101292
• CD Aliprantis, R. Tourky: Cones and duality , American Mathematical Society (2007), ISBN 0821841467