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