# Semi-ordered space

A common name for vector spaces on which there is defined a binary partial order relation that is compatible in a certain way with the vector space structure (cf. Vector space). The introduction of an order in function spaces makes it possible to study within the framework of functional analysis problems that are essentially connected with inequalities between functions. However, in contrast to the set of real numbers, which is totally ordered, the natural order in function spaces is only partial; for example, in the space it is natural to say that a function majorizes a function if for all . Under this definition of order, many functions are incomparable with each other.

## Ordered vector spaces.

A vector space over the field of real numbers is called ordered if there is defined on it a binary order relation , where implies for any and for any number . An example is with the natural order. If is an order, then the set is a cone, called the positive cone. Conversely, if in a certain space a cone with vertex at the origin is given, then can be given an order under which : one should put if . One considers also more general ordered vector spaces, in which only a quasi-order structure is defined. In this case is a wedge, and every wedge with vertex at the origin generates a quasi-order in (cf. also Wedge (in a vector space)).

Suppose that a vector space has been provided with an order. The cone is called generating if . This property of is necessary and sufficient for any finite subset of to be bounded (from above and below). The ordered vector spaces in which every set bounded from above has a least upper bound, or supremum, and hence also every set bounded from below has a greatest lower bound, or infimum, are called order complete or (o)-complete. A weaker type of completeness for ordered vector spaces is defined as follows: An ordered vector space is called Dedekind complete if every set which is bounded from above and directed upwards has a least upper bound (a set is directed upwards if for any there is an such that ; cf. also Directed set). If this requirement is satisfied for bounded increasing sequences, then the ordered vector space is called Dedekind (o)-complete. Dedekind completeness is weaker than (o)-completeness. For example, if is an arbitrary infinite-dimensional Banach space, , , and if is the cone spanned by the closed ball and the element , and is given the order using , then is Dedekind complete but not (o)-complete. An ordered vector space is called Archimedean if the Archimedean axiom holds in it. In particular, every Dedekind (o)-complete ordered vector space is Archimedean.

One introduces in an ordered vector space the notion of order convergence: A sequence (o)-converges to an element if there are increasing and decreasing sequences and for which and . The (o)-limit has many of the properties of the limit in the set of real numbers, although some of these hold only in Archimedean ordered vector spaces.

A linear operator mapping the ordered vector space to an ordered vector space (in particular, a real-valued linear functional) is called positive if . For positive functionals there is the following theorem on extensions. Let be a linear subset of which majorizes the cone (this means that for any there is a with ). Then every linear functional given on and positive with respect to the cone admits a linear positive extension to all of .

## Vector lattices

A vector lattice is an ordered vector space in which the order relation defines a lattice structure. Here, for the definition of a vector lattice it suffices to postulate the existence of one of the bounds: the upper or the lower , for any two elements of the space. For example, if exists, then . If is a vector lattice, then the cone is called minihedral. In a vector lattice, for any element its positive and negative parts exist: and . Here , and this formula gives the "minimal" representation of as a difference of positive elements, that is, if , where , then , . A minihedral cone is generating. The element is called the modulus of the element . In the space with the natural order, the positive cone is minihedral, the positive part of any function is obtained from by replacing its negative values by zero, while the modulus is the function . In a vector lattice, every finite set of elements has upper and lower bounds. The modulus of an element in a vector lattice has many of the properties of the absolute value of a real number.

The lattice of a vector lattice is distributive. In fact, it satisfies the stronger condition: For an arbitrary set of its elements for which exists, the following formula holds for any : . Then the dual formula also holds (cf. also Distributive lattice).

The theorem on the double partition of positive elements follows: If , where , and simultaneously , where all the , then every can be represented as in such a way that and

Two elements in a vector lattice are called disjoint if . Two sets are called disjoint if for any , . In the space the disjointness means that . A positive element is called a weak unit (a unit in the sense of Freudenthal) if 0 is the only element disjoint to . In , any function which is greater than 0 on an everywhere-dense set is a weak unit. However, if an element is such that for any there exists a for which , then is called a strong unit, and an with a strong unit is called a vector lattice of bounded elements. In , any function for which is a strong unit. If in an Archimedean vector lattice with a strong unit one puts , then becomes a normed lattice.

In the plane, any cone, apart from a one-dimensional cone (that is, a ray), is minihedral. However, in higher-dimensional spaces there are many non-minihedral closed cones, for example all "circular" cones in . For a cone (with vertex at zero) in an -dimensional Archimedean ordered vector space to be minihedral it is necessary and sufficient that it should be spanned by an -dimensional simplex with linearly independent vertices. Every Archimedean -dimensional vector lattice is isomorphic to the space with the coordinatewise ordering.

## $K$-spaces.

Also called Kantorovich spaces. These are (o)-complete vector lattices. This is the main class of semi-ordered spaces; they are always Archimedean. The notion of (o)-convergence in a -space is described in terms of upper and lower limits; namely, for a bounded sequence ,

and then means that . Let be a -space. For any set , the set is called its disjoint complement. A set which is the disjoint complement of another set is called a band. For any set there is a smallest band containing , namely ; it is called the band generated by the set . If itself is a band, then . The band generated by a singleton set is called principal. The notion of a band is also introduced in any vector lattice; however, in a -space it plays a special role, since one has the following theorem on projecting onto a band: If is a band in , then for any there exists a unique decomposition , where , . The linear operator defined here is called projection onto the band . If one is given an arbitrary collection of pairwise disjoint bands , complete in the sense that 0 is the only element of disjoint from all the , then any can be written as , where . Every lattice-ideal (-ideal) is also a -space. However, if and in , then this relation also holds in only in the case when the sequence is bounded in .

An example of a -space is the space of all real-valued almost-everywhere finite measurable functions on , in which equivalent functions are identified. A function is assumed to be positive if almost-everywhere. If is a countable subset of that is bounded from above (being bounded from above means that there is a such that almost-everywhere for any ), then the function will be the least upper bound of the set , that is, can be computed pointwise. However, for uncountable sets, the calculation of bounds in this way is already impossible, and for uncountable sets it is more difficult to establish the existence of least upper bounds in . In , (o)-convergence means convergence almost-everywhere. All the spaces , , are lattice-ideals in , and hence are also -spaces.

An important role is played by the Riesz–Kantorovich theorem, stating that the set of all order-bounded operators (that is, linear operators taking order-bounded sets to order-bounded sets) from a vector lattice into a -space with the natural order ( means that for all ) is itself a -space. The theory of -spaces has found applications in convex analysis and in the theory of extremum problems. Many results here are based on the Hahn–Banach–Kantorovich theorem on the extension of linear operators with values in a -space.

A -space is called extended (or laterally complete) if every set of pairwise disjoint elements in it is bounded. An extended -space always has a weak unit. For any -space , there exists a unique (up to an isomorphism) extended -space in which is imbedded as an -ideal, and the band in generated by coincides with . Such a is called the maximal extension of the -space . The space is the maximal extension of all the spaces . The notion of an extended -space plays an important role in the theory of semi-ordered spaces, in particular in representing a -space by functions.

Closely associated with a vector lattice and a -space is the notion of a lattice-normed space — a vector space to each element of which corresponds its generalized norm, which is an element of a fixed vector lattice and which satisfies the usual norm axioms, in which the inequality sign is understood in the sense of the order of the given vector lattice. Such spaces are used in the theory of functional equations (existence theorems; methods for approximate solution; the Newton–Kantorovich method; monotone processes of successive approximation, etc.).

## Topological semi-ordered spaces.

In functional analysis one also uses ordered vector spaces on which there is also defined a certain topology compatible with the order. The simplest and most important example of such a space is a Banach lattice. A generalization of the concept of a Banach lattice is that of a locally convex lattice.

An important class of Banach -spaces consists of the Kantorovich–Banach spaces, or -spaces. This is a Banach -space satisfying two additional conditions: 1) implies (order-continuity of the norm); 2) if the sequence is increasing and not order-bounded, then . In -spaces it is possible to describe in terms of the norm many facts the meaning of which depend only on the order. For example, means that as , uniformly with respect to . For a set in a -space to be order-bounded it is necessary and sufficient that the set of all numbers of the form , where , should be bounded. A -space is a regular -space.

An example of a -space: for .

Let be an arbitrary locally convex space equipped with an ordered vector space structure and having a so-called normal cone ; here normality of is equivalent to the supposition that has a base of absolutely-convex and order-saturated neighbourhoods of zero (meaning that if and , then also the whole interval ). For every continuous linear functional on a locally convex ordered vector space to be representable as the difference of positive continuous linear functionals, it is necessary and sufficient that the cone is normal in the weak topology. For normed spaces, normality of the cone in the weak and in the strong topology are equivalent.

#### References

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