Cones and Duality
Ordered vector spaces and cones were introduced in mathematics at the beginning of the 20th century and were developed in parallel with functional analysis and operator theory. Since cones are employed to solve optimization problems, the theory of ordered vector spaces is an indispensable tool for solving a variety of applied problems appearing in areas such as engineering, econometrics and the social sciences.
The aim of this book is to present the theory of ordered vector spaces from a contemporary perspective, which has been influenced by a study of ordered vector spaces in economics as well as other recent applications. The material is spread out in eight chapters. The first chapters start with fundamental properties of wedges and cones and illustrate a variety of remarkable results from the connection between the topology and the order. The role of the Riesz decomposition property and normal cones is pointed out. The next section studies in detail cones in finite dimensional spaces and polyhedral cones. The authors proceed with a presentation of the fixed points and eigenvalues of Krein operators. Further chapters contain material on K-lattices, Riesz-Kantorovich functionals and piecewise affine functions, which is a topic that has not been included in any monograph before. The last chapter serves as an appendix on linear topologies and their basic properties.
At the end of each section, there is a list of exercises of varying degrees of difficulty designed to help the reader to better understand the material. This aim is further supported by the provision of hints to selected exercises. Since the topics discussed in the book have their origins in problems from economics and finance, the book will be valuable not only for students and researchers in mathematics but also for those interested in economics, finance and engineering.