Chapter 1

Cones

A wedge is any nonempty subset of a vector space that is closed under vector

addition and scalar multiplication by nonnegative real numbers. A cone is

a wedge that does not contain any negative multiple of its nonzero vectors.

In every ordered vector space the set of vectors that dominate zero is a cone

called the positive cone. Conversely, every cone induces a vector ordering on

the vector space in which it is the positive cone. Therefore, cones and vector

orderings are related in a one-to-one manner. In other words, an ordered

vector space is a vector space equipped with cone.

This chapter studies the basic properties of cones. Like the rest of this

book, the chapter talks about cones in the language of ordered vector spaces.

The emphasis is to articulate features related to the order structure of an

ordered vector space in terms of the geometry of the positive cone and

to translate geometric properties of cones back into the properties of the

vector ordering. The material in the chapter is presented without explicitly

invoking any topological concepts.

In the first section, we introduce wedges and cones. In the second section,

we study the special class of Archimedean cones. The important class of

lattice cones (cones of vector lattices) is discussed in Section 1.3 and the basic

lattice identities of the lattice cones are stated and proved here. Sections 1.4,

1.5, 1.6, and 1.7 introduce and investigate order bounded operators, positive

linear functionals, cone bases, faces, and extremal rays.

The chapter concludes with a thorough discussion of the decomposition

and interpolation properties of cones in vector spaces. These important

properties are central to duality theory and are shared by lattice cones as

well as certain function spaces. These properties provide a rich duality

theory that can be described without invoking topological notions. The

1

http://dx.doi.org/10.1090/gsm/084/01