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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842601.png" /> it is natural to say that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842602.png" /> majorizes a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842603.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842604.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842605.png" />. Under this definition of order, many functions are incomparable with each other.

Ordered vector spaces.

A vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842606.png" /> over the field of real numbers is called ordered if there is defined on it a binary order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842607.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842608.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s0842609.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426011.png" /> for any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426012.png" />. An example is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426013.png" /> with the natural order. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426014.png" /> is an order, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426015.png" /> is a cone, called the positive cone. Conversely, if in a certain space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426016.png" /> a cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426017.png" /> with vertex at the origin is given, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426018.png" /> can be given an order under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426019.png" />: one should put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426021.png" />. One considers also more general ordered vector spaces, in which only a quasi-order structure is defined. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426022.png" /> is a wedge, and every wedge with vertex at the origin generates a quasi-order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426023.png" /> (cf. also Wedge (in a vector space)).

Suppose that a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426024.png" /> has been provided with an order. The cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426025.png" /> is called generating if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426026.png" />. This property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426027.png" /> is necessary and sufficient for any finite subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426028.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426029.png" /> is directed upwards if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426030.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426032.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426033.png" /> is an arbitrary infinite-dimensional Banach space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426035.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426036.png" /> is the cone spanned by the closed ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426037.png" /> and the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426039.png" /> is given the order using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426041.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426042.png" /> (o)-converges to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426043.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426044.png" /> if there are increasing and decreasing sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426046.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426048.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426049.png" /> mapping the ordered vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426050.png" /> to an ordered vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426051.png" /> (in particular, a real-valued linear functional) is called positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426052.png" />. For positive functionals there is the following theorem on extensions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426053.png" /> be a linear subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426054.png" /> which majorizes the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426055.png" /> (this means that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426056.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426057.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426058.png" />). Then every linear functional given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426059.png" /> and positive with respect to the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426060.png" /> admits a linear positive extension to all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426061.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426062.png" /> or the lower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426063.png" />, for any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426064.png" /> of the space. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426065.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426066.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426067.png" /> is a vector lattice, then the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426068.png" /> is called minihedral. In a vector lattice, for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426069.png" /> its positive and negative parts exist: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426071.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426072.png" />, and this formula gives the "minimal" representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426073.png" /> as a difference of positive elements, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426074.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426077.png" />. A minihedral cone is generating. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426078.png" /> is called the modulus of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426079.png" />. In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426080.png" /> with the natural order, the positive cone is minihedral, the positive part of any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426081.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426082.png" /> by replacing its negative values by zero, while the modulus is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426083.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426084.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426085.png" /> exists, the following formula holds for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426086.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426087.png" />. Then the dual formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426088.png" /> also holds (cf. also Distributive lattice).

The theorem on the double partition of positive elements follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426089.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426090.png" />, and simultaneously <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426091.png" />, where all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426092.png" />, then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426093.png" /> can be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426094.png" /> in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426095.png" /> and

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426096.png" />

Two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426097.png" /> in a vector lattice are called disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426098.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s08426099.png" />. Two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260100.png" /> are called disjoint if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260101.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260103.png" />. In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260104.png" /> the disjointness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260105.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260106.png" />. A positive element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260107.png" /> is called a weak unit (a unit in the sense of Freudenthal) if 0 is the only element disjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260108.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260109.png" />, any function which is greater than 0 on an everywhere-dense set is a weak unit. However, if an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260110.png" /> is such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260111.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260112.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260113.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260114.png" /> is called a strong unit, and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260115.png" /> with a strong unit is called a vector lattice of bounded elements. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260116.png" />, any function for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260117.png" /> is a strong unit. If in an Archimedean vector lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260118.png" /> with a strong unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260119.png" /> one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260120.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260121.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260122.png" />. For a cone (with vertex at zero) in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260123.png" />-dimensional Archimedean ordered vector space to be minihedral it is necessary and sufficient that it should be spanned by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260124.png" />-dimensional simplex with linearly independent vertices. Every Archimedean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260125.png" />-dimensional vector lattice is isomorphic to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260126.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260128.png" />-space is described in terms of upper and lower limits; namely, for a bounded sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260129.png" />,

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260130.png" />

and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260131.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260132.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260133.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260134.png" />-space. For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260135.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260136.png" /> is called its disjoint complement. A set which is the disjoint complement of another set is called a band. For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260139.png" /> there is a smallest band containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260140.png" />, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260141.png" />; it is called the band generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260143.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260144.png" /> itself is a band, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260145.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260147.png" />-space it plays a special role, since one has the following theorem on projecting onto a band: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260149.png" /> is a band in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260150.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260151.png" /> there exists a unique decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260152.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260154.png" />. The linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260155.png" /> defined here is called projection onto the band <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260157.png" />. If one is given an arbitrary collection of pairwise disjoint bands <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260158.png" />, complete in the sense that 0 is the only element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260159.png" /> disjoint from all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260160.png" />, then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260161.png" /> can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260162.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260163.png" />. Every lattice-ideal (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260164.png" />-ideal) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260165.png" /> is also a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260166.png" />-space. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260168.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260169.png" />, then this relation also holds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260170.png" /> only in the case when the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260171.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260172.png" />.

An example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260173.png" />-space is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260174.png" /> of all real-valued almost-everywhere finite measurable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260175.png" />, in which equivalent functions are identified. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260176.png" /> is assumed to be positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260177.png" /> almost-everywhere. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260178.png" /> is a countable subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260179.png" /> that is bounded from above (being bounded from above means that there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260180.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260181.png" /> almost-everywhere for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260182.png" />), then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260183.png" /> will be the least upper bound of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260184.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260185.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260186.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260187.png" />, (o)-convergence means convergence almost-everywhere. All the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260189.png" />, are lattice-ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260190.png" />, and hence are also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260191.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260192.png" />-space with the natural order (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260193.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260194.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260195.png" />) is itself a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260196.png" />-space. The theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260197.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260198.png" />-space.

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260199.png" />-space is called extended (or laterally complete) if every set of pairwise disjoint elements in it is bounded. An extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260202.png" />-space always has a weak unit. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260203.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260204.png" />, there exists a unique (up to an isomorphism) extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260205.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260206.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260207.png" /> is imbedded as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260208.png" />-ideal, and the band in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260209.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260210.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260211.png" />. Such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260212.png" /> is called the maximal extension of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260214.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260215.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260216.png" /> is the maximal extension of all the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260217.png" />. The notion of an extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260218.png" />-space plays an important role in the theory of semi-ordered spaces, in particular in representing a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260219.png" />-space by functions.

Closely associated with a vector lattice and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260220.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260221.png" />-spaces consists of the Kantorovich–Banach spaces, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260223.png" />-spaces. This is a Banach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260224.png" />-space satisfying two additional conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260225.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260226.png" /> (order-continuity of the norm); 2) if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260227.png" /> is increasing and not order-bounded, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260228.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260229.png" />-spaces it is possible to describe in terms of the norm many facts the meaning of which depend only on the order. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260230.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260231.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260232.png" />, uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260233.png" />. For a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260234.png" /> in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260235.png" />-space to be order-bounded it is necessary and sufficient that the set of all numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260236.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260237.png" />, should be bounded. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260238.png" />-space is a regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260239.png" />-space.

An example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260241.png" />-space: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260242.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260243.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260244.png" /> be an arbitrary locally convex space equipped with an ordered vector space structure and having a so-called normal cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260245.png" />; here normality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260246.png" /> is equivalent to the supposition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260247.png" /> has a base of absolutely-convex and order-saturated neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260248.png" /> of zero (meaning that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260249.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260250.png" />, then also the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260251.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084260/s084260252.png" /> is normal in the weak topology. For normed spaces, normality of the cone in the weak and in the strong topology are equivalent.

References

[1]	B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff (1967) (Translated from Russian)
[2]	L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker, "Functional analysis in semi-ordered spaces" , Moscow-Leningrad (1950) (In Russian)
[3]	H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[4]	M.A. Krasnosel'skii, "Positive solutions of operator equations" , Wolters-Noordhoff (1964) (Translated from Russian)
[5]	M.Ya. Antonovskii, V.G. Boltyanskii, T.A. Sarymsakov, "Topological Boolean algebras" , Tashkent (1963) (In Russian)
[6]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[7]	L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)
[8]	, Functional analysis , Math. Reference Library , Moscow (1972) (In Russian)
[9]	B.Z. Vulikh, "Introduction to the theory of cones in normed spaces" , Kalinin (1977) (In Russian)
[10]	M.G. Krein, M.A. Rutman, "Linear operators leaving invariant a cone in a Banach space" Transl. Amer. Math. Soc. , 26 (1956) Uspekhi Mat. Nauk , 3 : 1 (1948) pp. 3–95
[11]	A.V. Bukhvalov, A.I. Veksler, G.Ya. Lozanovskii, "Banach lattices - some Banach aspects of their theory" Russian Math. Surveys , 34 : 2 (1979) pp. 159–212 Uspekhi Mat. Nauk , 34 : 2 (1979) pp. 137–183
[12]	G.P. Akilov, S.S. Kutateladze, "Ordered vector spaces" , Novosibirsk (1978) (In Russian)

Comments

Cf. also Riesz space.

References

[a1]	H. Freudenthal, "Teilweise geordnete Moduln" Proc. Royal Acad. Sci. Amsterdam , 39 (1936) pp. 641–651
[a2]	H.H. Schaefer, "Banach lattices and positive operators" , Springer (1974)
[a3]	W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
[a4]	A.C. Zaanen, "Riesz spaces" , II , North-Holland (1982)