Ordered vector space

An ordered vector space is a mathematical structure. It is a vector space on which a compatible with the vector space structure ordering relation is given in addition that is usually referred to (it reads less than or equal ). This makes it possible to compare the elements of a vector space of size. Many investigated in mathematics vector spaces carry a natural order structure.

Definition

An ordered vector space is a pair such that the following holds consisting of a vector space and an order relation on:

• For all, ie is reflexive.
• From and it follows for all, ie is transitive.
• It follows for all, that is, with the addition tolerated.
• It follows for all and that is with the scalar multiplication tolerated.

In the definition you can by a parent body to replace. In most applications, however, one has to do it with the body of real numbers. A vector space is called ordered vector space if it is organized as a real vector space. Many of the discussed here conceptions can be generalized to ordered abelian groups.

Positive cone

Is an ordered vector space, ie, the positive cone. It is in fact a cone, that is, it is:

• For all and is.

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

Conversely, in a vector space, a cone is given, is defined by an ordering relation, which makes you a parent vector space, such that. An ordered vector space can therefore also be regarded as a vector space with an excellent bowling. Characteristics of order can be related to algebraic and geometric properties of the cone; is even a topological vector space, so topological properties of the cone will be added.

Positive operators

The structure- preserving mappings between ordered vector spaces and the linear operators that preserve the order structure, ie for which follows from always. Such maps are called positive or monotone operators. The examination of positive operators is an important part of the theory of ordered vector spaces.

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

An order interval is a set of the form. A linear operator between ordered vector spaces is called proper limited if it maps order intervals in order intervals. Differences of positive operators are apparently properly restricted.

Dual order

Is an ordered vector space, then a cone, which makes the dual space into an ordered vector space; This is due to the so-called dual order. Is also a topological vector space, is considered instead of the algebraic topological the dual space, that is the space of all continuous linear functionals. Is this normalized space, or more generally locally convex, so is the rich for this room classes duality theory available.

Often one considers only the subspace of proper limited Functional and speaks of the proper limited dual space.

Examples

• The sequence spaces as, or are ordered vector spaces, if the order component-wise stated, ie if one defined by two sequences and the relation.
• Function spaces like or Lp [0,1] are ordered vector spaces, if pointwise explains the order, that is, if one defines two functions and the relation through for all of the domain, or almost anywhere on the domain.
• Is a C *-algebra and sets one, then one can show that a cone is turning into an ordered vector space. The study of the dual space with the dual order is an important method in the theory of C * - algebras.

More Tendings

Be an ordered vector space.

Strict Rules

In the definition given here has not demanded that out and always follow is; the order relation would then be antisymmetric, and this would be equivalent to saying that the cone is pointed ( ie ). The most occurring in the applications cones are pointed. Some authors consider to be a cone is always a pointed cone and call the previously introduced general term an obtuse cone. Antisymmetric orders are also called strict orders.

Looking order

The order directed on to say, if it always admits two elements with and. The order is then directed exactly when, that is, if the positive cone generates the vector space.

Order units

An element is called an order unit, if one exists for every with. This is equivalent to saying that the order interval is an absorbing set.

Apparently, the constant function 1 is an order unit in, during the sequence space has no order units.

Archimedean order

The order called on Archimedean if: Sind and for all, as follows.

The order is called almost Archimedean if: Sind and for all, as follows.

The order is called nowhere Archimedean if there is one for every with for all.

Subspaces, quotients and direct products

Is an ordered vector space and a lower space, it is restricted to the order of an ordered vector space again, it is evident and the embedding is a positive operator.

The quotient space is the cone appears to be an ordered vector space and the quotient map is a positive operator.

Finally, if a family of ordered vector spaces, so the direct product becomes a parent vector space if one explains the positive cone 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.

Riesz spaces

A directed vector space has the Riesz'sche interpolation property if the following holds:

If and, then there's, and.

Are there any two elements of an ordered vector space is always a smallest element and which is then given and the supremum and is, as one speaks of a Riesz space or vector lattice. It can be shown that in fact a distributive lattice is present, wherein the other operation could be defined by association. Apparently vector associations have the Riesz'sche interpolation property, because the above definition can be satisfied with. This is called a vector lattice completely when not only two elements but each bounded above set has a supremum.

Remarks Title: Some authors call directed and strictly ordered vector spaces with Riesz interpolation property of Riesz'schen areas, see, for example, and therefore use Riesz space not as a synonym for vector lattice.

In connection with the terms introduced here the following important theorem of F. Riesz consists of:

• Is a directed and strictly ordered vector space with the Riesz'schen interpolation property, the order restricted dual space is a complete vector lattice.

As an application, consider a C * algebra. Then the self-adjoint part is a real vector space, which is through the cone to a directed and strictly ordered vector space with Riesz'scher interpolation property .. The dual space, which coincides with the proper restricted dual space, therefore, is a complete vector lattice, which for the C * - theory is of importance.

Topological ordered vector spaces

Carries an ordered vector space in addition a vector space topology, this is called an ordered topological vector space and can investigate continuity properties of order. In particular, in vector associations you can see the continuity of the pictures

Study.

We have the following theorem for ordered topological vector Associations:

• The mapping is continuous if a base of neighborhoods of quantities has that have the following property: If and with, as follows.

Even is a normed space with norm and a vector lattice, it is called the standard is a lattice norm if for always follows .. In this case it is called a normed vector lattice. Then, the above- quoted sentence is applicable and to recognize the continuity of the lattice operations. Typical examples are the examples listed above or with their natural orders and standards.

For ordered topological vector spaces, in particular ordered Banach spaces, there exists an extensive theory, it is referred for at this point in the literature.