Cones and duality / Charalambos D. Aliprantis, Rabee Tourky.

Format:

Book

Author/Creator:

Aliprantis, Charalambos D., author.

Tourky, Rabee, author.

Series:

Graduate studies in mathematics ; Volume 84.

Graduate Studies in Mathematics ; Volume 84

Language:

English

Subjects (All):

Cones (Operator theory).

Linear topological spaces, Ordered.

Physical Description:

1 online resource (279 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2007.

Language Note:

English

Summary:

Ordered vector spaces and cones made their debut in mathematics at the beginning of the twentieth century. They were developed in parallel (but from a different perspective) with functional analysis and operator theory. Before the 1950s, ordered vector spaces appeared in the literature in a fragmented way. Their systematic study began around the world after 1950 mainly through the efforts of the Russian, Japanese, German, and Dutch schools. Since cones are being employed to solveoptimization problems, the theory of ordered vector spaces is an indispensable tool for solving a variety of applied problems appearing in several diverse areas, such as engineering, econometrics, and the social sciences. For this reason this theory plays a prominent role not only in functionalanalysis but also in a wide range of applications. This is a book about a modern perspective on cones and ordered vector spaces. It includes material that has not been presented earlier in a monograph or a textbook. With many exercises of varying degrees of difficulty, the book is suitable for graduate courses. Most of the new topics currently discussed in the book have their origins in problems from economics and finance. Therefore, the book will be valuable to any researcher and graduatestudent who works in mathematics, engineering, economics, finance, and any other field that uses optimization techniques.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""The ""isomorphism"" notion""; ""Chapter 1. Cones""; ""Â1.1. Wedges and cones""; ""Â1.2. Archimedean cones""; ""Â1.3. Lattice cones""; ""Â1.4. Positive and order bounded operators""; ""Â1.5. Positive linear functionals""; ""Â1.6. Faces and extremal vectors of cones""; ""Â1.7. Cone bases""; ""Â1.8. Decomposability in ordered vector spaces""; ""Â1.9. The Rieszâ€?Kantorovich formulas""; ""Chapter 2. Cones in topological vector spaces""; ""Â2.1. Ordered topological vector spaces""; ""Â2.2. Wedge duality""

""Â2.3. Normal cones""""Â2.4. Positivity and continuity""; ""Â2.5. Ordered Banach spaces""; ""Â2.6. Ice cream cones in normed spaces""; ""Â2.7. Ideals in ordered vector spaces""; ""Â2.8. The order topology generated by a cone""; ""Chapter 3. Yudin and pull-back cones""; ""Â3.1. Closed cones in finite dimensional vector spaces""; ""Â3.2. Directional wedges and Yudin cones""; ""Â3.3. Polyhedral wedges and cones""; ""Â3.4. The geometrical structure of polyhedral cones""; ""Â3.5. Linear inequalities and alternatives""; ""Â3.6. Pull-back cones of operators""

""Chapter 4. Krein operators""""Â4.1. The concept of a Krein operator""; ""Â4.2. Eigenvalues of Krein operators""; ""Â4.3. Fixed points and eigenvectors""; ""Chapter 5. K-lattices""; ""Â5.1. The notion and properties of K-lattices""; ""Â5.2. The Rieszâ€?Kantorovich transform""; ""Â5.3. The order extension of L[sub(b)](L, N)""; ""Chapter 6. The order extension of L'""; ""Â6.1. The extension of L'""; ""Â6.2. Generalized Rieszâ€?Kantorovich functionals""; ""Â6.3. When is the Rieszâ€?Kantorovich functional additive?""; ""Chapter 7. Piecewise affine functions""

""Â7.1. One-dimensional piecewise affine functions""""Â7.2. Multivariate piecewise affine functions""; ""Chapter 8. Appendix: linear topologies""; ""Â8.1. Linear topologies on vector spaces""; ""Â8.2. Duality theory""; ""Â8.3. G-topologies""; ""Â8.4. The separation of convex sets""; ""Â8.5. Normed and Banach spaces""; ""Â8.6. Finite dimensional topological vector spaces""; ""Â8.7. The open mapping and the closed graph theorems""; ""Â8.8. The bounded weak* topology""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""

""M""""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""X""; ""Y""; ""Back Cover""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-2114-3