Preface

XI

Chapters 5, 6, and 7 contain new material that, as far as we know,

has not appeared before in any monograph. Chapter 5 develops in detail

the theory of /C-lattices. An ordered vector space L is called a fC-lattice,

where K is a super cone of L, i.e., K I) L

+

, if for every nonempty subset A

of L the collection of all L+-upper bounds of A is nonempty and has a /C-

infimum. As can be seen immediately from this definition, the notion of a /C-

lattice has applications to optimization theory. Chapter 5 also introduces the

notion of the Riesz-Kantorovich transform for an m-tuple of order bounded

operators that is used to investigate the fundamental duality properties of

ordered bounded operators from an ordered vector space to a Dedekind

complete Riesz space. Subsequently, using the theory of/C-lattices, we define

an important order extension of £&(L, iV), the ordered vector space of all

order bounded operators from L to a Dedekind complete Riesz space. This

extension allows us to enrich the lattice structure of Cb{L,N) in a useful

manner.

Chapter 6 specializes the theory of /C-lattices to the space of all ordered

bounded linear functional on an ordered vector space. Among other things,

this chapter introduces an important order extension of V called the "super

topological dual of I/" and studies its fundamental properties. Moreover,

in this chapter the reader will find several interesting optimization results.

In essence, Chapter 6 brings, via the concept of a /C-lattice, the theory of

ordered vector spaces to the theory of linear minimization. In other words,

this chapter can be viewed as contributing new functional analytic tools to

the study of linear minimization problems.

Finally, in Chapter 7 we present a comprehensive investigation of piece-

wise affine functions. It turns out that their structure is intimately related

to order and lattice properties that are discussed in detail in this chapter.

The main result here is that the collection of piecewise affine functions co-

incides with the Riesz subspace generated by the affine functions. Piecewise

affine (or piecewise linear) functions are very important in approximation

theory and have been studied extensively in one-dimensional settings. How-

ever, even until now, in dimensions more than one there seems to be no

satisfactory theory of piecewise affine functions. They are defined on finite

dimensional spaces, and no attempt has been made to generalize their theory

to infinite dimensional settings. This provides the opportunity for several

future research directions.

At the end of each section there is a list of exercises of varying degrees

of difficulty designed to help the reader comprehend the material in the

section. There are almost three hundred and fifty exercises in the book.

Hints to selected exercises are also given. The inclusion of the exercises

makes the book, on one hand, suitable for graduate courses and, on the