**Graduate Studies in Mathematics**

Volume: 84;
2007;
279 pp;
Hardcover

MSC: Primary 46; 47;
Secondary 06; 28; 91

Print ISBN: 978-0-8218-4146-4

Product Code: GSM/84

List Price: $60.00

Individual Member Price: $48.00

**Electronic ISBN: 978-1-4704-2114-4
Product Code: GSM/84.E**

List Price: $60.00

Individual Member Price: $48.00

#### Supplemental Materials

# Cones and Duality

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*Charalambos D. Aliprantis; Rabee Tourky*

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 solve optimization 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 functional analysis 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 graduate student who works in mathematics,
engineering, economics, finance, and any other field that uses
optimization techniques.

#### Table of Contents

# Table of Contents

## Cones and Duality

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- The "isomorphism" notion xiii14 free
- Chapter 1. Cones 116 free
- §1.1. Wedges and cones 217
- §1.2. Archimedean cones 1126
- §1.3. Lattice cones 1328
- §1.4. Positive and order bounded operators 2338
- §1.5. Positive linear functionals 3146
- §1.6. Faces and extremal vectors of cones 3651
- §1.7. Cone bases 3954
- §1.8. Decomposability in ordered vector spaces 4358
- §1.9. The Riesz–Kantorovich formulas 5469

- Chapter 2. Cones in topological vector spaces 6176
- §2.1. Ordered topological vector spaces 6277
- §2.2. Wedge duality 7085
- §2.3. Normal cones 7691
- §2.4. Positivity and continuity 8297
- §2.5. Ordered Banach spaces 85100
- §2.6. Ice cream cones in normed spaces 99114
- §2.7. Ideals in ordered vector spaces 103118
- §2.8. The order topology generated by a cone 110125

- Chapter 3. Yudin and pull-back cones 117132
- §3.1. Closed cones in finite dimensional vector spaces 118133
- §3.2. Directional wedges and Yudin cones 122137
- §3.3. Polyhedral wedges and cones 131146
- §3.4. The geometrical structure of polyhedral cones 137152
- §3.5. Linear inequalities and alternatives 148163
- §3.6. Pull-back cones of operators 152167

- Chapter 4. Krein operators 159174
- Chapter 5. K-lattices 173188
- Chapter 6. The order extension of L' 197212
- Chapter 7. Piecewise affine functions 221236
- Chapter 8. Appendix: linear topologies 243258
- §8.1. Linear topologies on vector spaces 244259
- §8.2. Duality theory 247262
- §8.3. G-topologies 249264
- §8.4. The separation of convex sets 251266
- §8.5. Normed and Banach spaces 252267
- §8.6. Finite dimensional topological vector spaces 256271
- §8.7. The open mapping and the closed graph theorems 257272
- §8.8. The bounded weak* topology 259274

- Bibliography 265280
- Index 271286
- Back Cover Back Cover1298

#### Readership

Graduate students and research mathematicians interested in functional analysis and applications, in particular to optimization.

#### Reviews

...the book will be valuable not only for students and researchers in mathematics but also for those interested in economics, finance and engineering.

-- EMS Newsletter

As a whole, this book is an excellent reference of general interest in ordered vector spaces.

-- Mathematical Reviews

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.

-- Zentralblatt MATH