Cones and duality.

*(English)*Zbl 1127.46002
Graduate Studies in Mathematics 84. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4146-4/hbk). xiv, 279 p. (2007).

This is an introduction into the theory of ordered vector spaces which is oriented mainly towards the reader interested in applications of order, primarily, to economics. The aims of the book are well reflected in the authors’ dedication to Leonid Kantorovich (1912–1986).

The book consists of eight chapters. Chapter 1 deals with preliminaries about wedges and cones with an emphasis on decomposition and Riesz–Kantorovich formulas. Chapter 2 covers the basics of order ideals and topologies. Chapter 3 deals primarily with the so-called Yudin or minihedral cones in finite dimensions. Chapter 4 focuses on the fixed points and eigenvalues of Krein operators. In Chapters 5 and 6 the authors present the theory of \(\mathcal K\)-lattices. A \(\mathcal K\)-lattice is a vector space \(L\) with positive cone \(L_+\) and a supercone \(\mathcal K\) of \(L\) such that for every nonempty subset \(A\) of \(L\), the set of \(L_+\)-upper bounds of \(A\) is nonempty and has a \(\mathcal K\)-infimum. Chapter 7 treats piecewise affine functions in terms of ordered vector spaces.

This book will find its grateful readership as it bridges the gap between the theory of ordered vector spaces as cultivated in functional analysis and the theory of positivity as requested in applications to economics.

The book consists of eight chapters. Chapter 1 deals with preliminaries about wedges and cones with an emphasis on decomposition and Riesz–Kantorovich formulas. Chapter 2 covers the basics of order ideals and topologies. Chapter 3 deals primarily with the so-called Yudin or minihedral cones in finite dimensions. Chapter 4 focuses on the fixed points and eigenvalues of Krein operators. In Chapters 5 and 6 the authors present the theory of \(\mathcal K\)-lattices. A \(\mathcal K\)-lattice is a vector space \(L\) with positive cone \(L_+\) and a supercone \(\mathcal K\) of \(L\) such that for every nonempty subset \(A\) of \(L\), the set of \(L_+\)-upper bounds of \(A\) is nonempty and has a \(\mathcal K\)-infimum. Chapter 7 treats piecewise affine functions in terms of ordered vector spaces.

This book will find its grateful readership as it bridges the gap between the theory of ordered vector spaces as cultivated in functional analysis and the theory of positivity as requested in applications to economics.

Reviewer: S. S. Kutateladze (Novosibirsk)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46A40 | Ordered topological linear spaces, vector lattices |