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
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