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
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