Cone s an d Dualit y

http://dx.doi.org/10.1090/gsm/084

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Cone s an d Dualit y

Charalambo s D. Alipranti s RabeeTourk y

Graduate Studies

in Mathematics

Volum e 84

•& I p ^ S n l America n Mathematica l Societ y *0||jjO ? p r o v j c j e n c e i Rhod e Islan d

Editorial Board

David Cox (Chair) Walter Craig N. V. Ivanov

Steven G. Krantz

2000 Mathematics Subject Classification. P r i m a r y 46A40, 46B40, 47B60, 47B65; Secondary 06F30, 28A33, 91B28, 91B99.

For addi t ional information and upda t e s on th is book, visit w w w . a m s . o r g / b o o k p a g e s / g s m - 8 4

Library of Congress Cataloging-in-Publication D a t a

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

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 84) Includes bibliographical references and index. ISBN 978-0-8218-4146-4 (alk. paper) 1. Cones (Operator theory). 2. Linear topological spaces, Ordered. I. Tourky, Rabee, 1966-

II. Title.

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10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07

To the great Russian mathematician and economist

Leonid Vitaliyevich Kantorovich (1912-1986),

the 1975 Nobel Prize co-recipient in economics,

. . . whose brilliant ideas have shaped the field of ordered vector spaces and are present throughout this book.

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Contents

Preface

The "isomorphism" notion

Chapter 1. Cones

§1.1. Wedges and cones

§1.2. Archimedean cones

§1.3. Lattice cones

§1.4. Positive and order bounded operators

§1.5. Positive linear functionals

§1.6. Faces and extremal vectors of cones

§1.7. Cone bases

§1.8. Decomposability in ordered vector spaces

§1.9. The Riesz-Kantorovich formulas

Chapter 2. Cones in topological vector spaces

§2.1. Ordered topological vector spaces

§2.2. Wedge duality

§2.3. Normal cones

§2.4. Positivity and continuity

§2.5. Ordered Banach spaces

§2.6. Ice cream cones in normed spaces

§2.7. Ideals in ordered vector spaces

§2.8. The order topology generated by a cone

V l l l Contents

Chapter 3. Yudin and pull-back cones 117

§3.1. Closed cones in finite dimensional vector spaces 118

§3.2. Directional wedges and Yudin cones 122

§3.3. Polyhedral wedges and cones 131

§3.4. The geometrical structure of polyhedral cones 137

§3.5. Linear inequalities and alternatives 148

§3.6. Pull-back cones of operators 152

Chapter 4. Krein operators 159

§4.1. The concept of a Krein operator 160

§4.2. Eigenvalues of Krein operators 163

§4.3. Fixed points and eigenvectors 167

Chapter 5. K-lattices 173

§5.1. The notion and properties of K-lattices 174

§5.2. The Riesz–Kantorovich transform 183

§5.3. The order extension of £b(L, N) 190

Chapter 6. The order extension of V 197

§6.1. The extension of V 199

§6.2. Generalized Riesz-Kantorovich functionals 204

§6.3. When is the Riesz-Kantorovich functional additive? 210

Chapter 7. Piecewise affine functions 221

§7.1. One-dimensional piecewise affine functions 221

§7.2. Multivariate piecewise affine functions 227

Chapter 8. Appendix: linear topologies 243

§8.1. Linear topologies on vector spaces 244

§8.2. Duality theory 247

§8.3. 6-topologies 249

§8.4. The separation of convex sets 251

§8.5. Normed and Banach spaces 252

§8.6. Finite dimensional topological vector spaces 256

§8.7. The open mapping and the closed graph theorems 257

§8.8. The bounded weak* topology 259

Bibliography 265

Index 271

Preface

Ordered vector spaces made their debut 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 in various schools around the world after the 1950s. We mention the Russian school (headed by L. V. Kantorovich and the Krein brothers), the Japanese school (headed by H. Nakano), the Ger­man school (headed by H. H. Schaefer), and the Dutch school (headed by A. C. Zaanen). At the same time several monographs dealing exclu­sively with ordered vector spaces appeared in the literature; see for in­stance [55, 56, 71, 75, 89, 91]. The special class of ordered vector spaces known as Riesz spaces or vector lattices has been studied more extensively; see the monographs [14, 15, 66, 68, 86, 88, 93].

The theory of ordered vector spaces plays a prominent role in functional analysis. It also contributes to a wide variety of applications and is an indispensable tool for studying a variety of problems in engineering and economics; see for instance [29, 31 , 35, 36, 38, 42, 47, 49, 54, 64, 65, 76]. The introduction of Riesz spaces and more broadly ordered vector spaces to economic theory has proved tremendously successful and has allowed researchers to answer difficult questions in general price equilibrium theory, economies with differential information, the theory of perfect competition, and incomplete assets economies.

The goal of this monograph is to present the theory of ordered vector spaces from a contemporary perspective that has been influenced by the study of ordered vector spaces in economics as well as other recent appli­cations. We try to imbue the narrative with geometric intuition, which is

IX

X Preface

in keeping with a long tradition in mathematical economics. We also ap­proach the subject with our own personal presentiment that the special class of Riesz spaces is somehow "perfect" and thus loosely conceive of general ordered vector spaces as "deviations" from this "perfection." The book also contains material that has not been published in a monograph form before. The study of this material was initially motivated by various problems in economics and econometrics.

The material is spread out in eight chapters. Chapter 8 is an Appendix and contains some basic notions of functional analysis. Special attention is paid to the properties of linear topologies and the separation of convex sets. The results in this chapter (some of which are presented with proofs) are used throughout the monograph without specific mention.

Chapter 1 presents the fundamental properties of wedges and cones. Here we discuss Archimedean cones, lattice cones, extremal vectors of cones, bases of cones, positive linear functionals and the important decomposabil-ity property of cones known as the Riesz decomposition property. Chapter 2 introduces cones in topological vector spaces. This chapter illustrates the variety of remarkable results that can be obtained when some link between the order and the topology is imposed. The most important interrelationship between a cone and a linear topology is known as normality. We discuss nor­mal cones in detail and obtain several characterizations. In normed spaces, the normality of the cone amounts to the norm boundedness of the order intervals generated by the cone. In Chapter 2 we also introduce ideals and present some of their useful order and topological properties.

Chapter 3 studies in detail cones in finite dimensional vector spaces. The results here are much sharper. For instance, as we shall see, every closed cone of a finite dimensional vector space is normal. The reader will find in this chapter a study (together with a geometrical description) of the polyhedral cones as well as a discussion of the properties of linear inequalities—including a proof of "the Principle of Linear Programming." The chapter culminates with a study of pull-back cones and establishes the following "universality" property of C[0,1]: every closed cone of a finite dimensional vector space is the pull-back cone of the cone of C[0,1] via a one-to-one operator from the space to C[0,1].

Chapter 4 investigates the fixed points and eigenvalues of an important class of positive operators known as Krein operators. A Krein space is an ordered Banach space having order units and a closed cone. A positive oper­ator T on a Krein space is a Krein operator if for any x > 0 the vector Tnx is an order unit for some n. Many integral operators are Krein operators. These operators possess some useful fixed points that are investigated in this chapter.

Preface X I

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 co­incides 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. How­ever, 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

Xl l Preface

other hand, a reference source on ordered vector spaces and cones. It is our hope (and belief) that this monograph will not only be beneficial to mathematicians but also to other scientists in many disciplines, theoretical and applied, as well.

We take this opportunity to thank our late colleague Yuri Abramovich for reading an early draft of the manuscript and making numerous sugges­tions and corrections. The help provided to us by Monique Florenzano dur­ing the writing of the book is greatly appreciated. Besides correcting several faulty proofs, she recommended many important structural changes that im­proved the exposition of the book. Special thanks are due to Grainne Begley, Daniela Puzzello and Francesco Ruscitti for reading the manuscript carefully and correcting numerous (mathematical and nonmathematical) mistakes. We express our appreciation to our graduate students Iryna Topolyan, for reading the entire manuscript and finding an infinite number of mistakes, and Qianru Qi, for her help with the drawing of certain figures in the mono­graph.

C D . Aliprantis acknowledges with many thanks the financial support he received from the National Science Foundation under grants SES-0128039, DMI-0122214, and DMS-0437210, and the DOD Grant ACI-0325846. R. Tourky acknowledges with many thanks the financial support he received from the Australian Research Council under grant A00103450.

C. D. ALIPRANTIS, West Lafayette, Indiana, USA R. TOURKY, Queensland, Brisbane, AUSTRALIA

January 2007

The "isomorphism" notion

A typical mathematical field is usually described by a class of sets that are endowed with a "structure" concept that characterizes the subject matter of the field. Schematically, a typical branch of study in mathematics consists of pairs (X, 6 ) , where X is a set and & is the "structure" imposed on the set X that is characteristic to the area. The structure & can be expressed in terms of algebraic or topological properties or a mixture of the two. Here are a few examples of mathematical areas.

(1) Groups: Here for the typical object (X, S) , the structure & repre­sents the algebraic structure on X given by the operation of mul­tiplication (x,y) i—• xy and of the inverse function x \-+ x~l.

(2) Topological spaces: Here for the typical object (X, ©), the structure & represents the topology of the set X.

(3) Vector spaces: Here for the typical object (X, (3), the structure & represents the algebraic structure imposed on X by means of the addition (x, y) \-^ x + y and the scalar multiplication (A, x) i—> Ax .

(4) Topological vector spaces: Here for the typical object (X, 6 ) , the structure 6 represents the mixture of the vector space structure of X and the topological structure of X that makes the algebraic operations of X continuous.

(5) Ordered vector spaces: Here for the typical object (X, 6 ) , the struc­ture & represents the algebraic structure of X together with the vector ordering on X.

Once one deals with the objects of a mathematical field, one would like to have a way of identifying two objects of the field that look "alike."

Xiii

XIV The "isomorphism" notion

This is done with the "isomorphism" concept. The idea is very simple: We say that two objects ( X L , 6 I ) and (^2,62) from a mathematical field are isomorphic if there exists a one-to-one surjective (i.e., onto) function f:Xi —> X2 (called an isomorphism) such that /(@i) = ©2- The last identity should be intrepreted in the sense that / "preserves" the structures 61 and &2 of the sets X\ and X2, respectively. For instance, if X\ and X2 are isomorphic vector spaces via the isomorphism / , then (besides / being one-to-one and onto) it also satisfies f(ax -\- (3y) — otf{x) + (3f(y) for all x j G l i and all scalars a and (3. Likewise, by saying that two ordered vector spaces X\ and X2 are isomorphic via / , we mean that / : X\ —• X2 is a one-to-one surjective (linear) operator such that f(x) > f(y) holds in X2 if and only if x > y holds in X\.

The word "isomorphism" is the English version of the Greek word LiiaoiAOp(j)Lcrn6s" which etymologically is the composition of the Greek words "LCTOS" (which means equal, even, the same) and "fjopfiri" (which means form, figure, shape, appearance, structure). So, when we say that two mathematical objects are "isomorphic," we simply express the fact that they have the same (or similar or identical) shape (or form or appearance) and the concept of an "isomorphism" simply designates the state of being "isomorphic."

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89. B. Z. Vulikh, Special Topics in Geometry of Cones in Normed Spaces, Kalinin State University, 1978.

90. H. Weyl, The elementary theory of convex polyhedra, in: H. W. Kuhn and A. W. Tucker, eds., Contributions to the Theory of Games I, Annals of Mathematics Studies, Vol. 24 (Princeton University Press, Princeton, NJ, 1950), pp. 3-18. [Translation by H. W. Kuhn of the original paper by H. Weyl: Elementare Theorie der konvexen Polyeder, Coment. Math. Helv. 7(1934-35), 290-306.]

91. Y. C. Wong and K. F. Ng, Partially Ordered Topological Vector Spaces, Clarendon Press, Oxford, 1973.

92. A. I. Yudin, A solution of two problems in the theory of partially ordered spaces, Doklady Akad. Nauk SSSR 23 (1939), 418-422.

93. A. C. Zaanen, Riesz Spaces II, North-Holland, Amsterdam, 1983.

94. G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, #152 , Springer-Verlag, New York and Heidelberg, 1995.

Index

A0*, one-sided bipolar of set A, 135, 249 A*, one-sided polar of set A, 134, 249 AfF, vector space of afRne functions, 227 A°, polar of set A, 248 A°°, bipolar of set A, 249 Ax, half-ray generated by x, 37, 40 absolute polar of a set, 248 absolute value

of functional, 56 of operator, 54 of vector, 14

absorbing set, 33, 110, 244 additive function, 23, 184 additive mapping, 23, 184 adjoint operator, 148

Hilbert space, 109 affine function, 221, 227 afRne transformation, 231 Alaoglu-Bourbaki theorem, 248 algebraic dual, 205, 246 algebraic dual of a vector space, 31, 246 algebraic separating hyperplane theorem,

251 almost Archimedean cone, 12 almost Archimedean ordered vector space,

12 alternative

concave, 150 Farkas, 149 operator, 148 Stiemke, 149 wedge, 148

Ando's theorem, 66, 88, 91, 92 angle of an ice cream cone, 99 antisymmetry, 3 Archimedean cone, 11, 63, 119, 128

Archimedean ordered vector space, 11 Archimedean property, 11 arrangement generated by afRne functions,

235 arrangement of hyperplanes, 234 associative law, 178 atom, 39 axis of ice cream cone, 99

/3(L, Z/), strong topology on L, 251 /3(Z/,L), strong topology on L', 251 A°°, bipolar of set A, 249 [£], full hull of set B, 6 balanced set, 6, 244 Banach lattice, 85 Banach space, 253

ordered, 85 reflexive, 254 separable, 254 with the Levi property, 89 with the strong Levi property, 89

Banach-Steinhaus theorem, 254 base of cone, 39

pseudo, 200 basis

Hamel, 124 positive, 125 Yudin, 125

bilinear mapping, 247 bilinearity of a dual pair, 247 bipolar of a set, 249 bipolar theorem, 249

one-sided, 135, 249 Birkhoff's identity, 21 Birkhoff's inequalities, 21 Borel sets, 75

271

272 Index

bornological space, 111 bornological topology, 111 bounded above subset, 6 bounded below subset, 6

C-decomposition property, 188 r-complete topological vector space, 245 ca(E), the vector space of all countably

additive measures on E, 21 Cauchy net, 245 cells generated by affine functions, 235 cells induced by a hyperplane arrangement,

235 characteristic pairs of piecewise affine

function, 228 characterization of piecewise affine

function, 230 circled hull of a set, 254 circled set, 6, 244 closed and generating cone, 85 closed ball of seminorm, 245 closed circled hull of a set, 254 closed cone, 63, 85, 118 closed convex hull

compact, 254 closed convex hull of a set, 254 closed graph theorem, 259 closed unit ball, 68, 85, 253

metrizable, 254 cofinal vector subspace, 9 compact operator, 161 compact set, 255 compatible topology, 73, 248 complete set, 245 components of piecewise affine function,

222, 228 composition operator, 170 comprehensive set, 32 concave alternative, 150 concave function, 182 cone, 2

/C-lattice, 177 cr-order complete, 109 almost Archimedean, 12 Archimedean, 11, 63, 119, 128 base, 39 closed, 63, 85, 118 closed and generating, 85, 121 continuous pull-back, 152 convex, 2 dominance, 152 dual of a cone, 70 extension, 191 extremal ray, 40 generating, 4 ice cream, 41, 99 lattice, 13, 14

lexicographic, 11 normal, 76 pointed convex, 2 pointed convex with vertex at zero, 2 polyhedral, 131 product, 8 pull-back, 152, 158 subcone, 37 super, 174 super topological dual, 199 with the decomposition property, 43 with the interpolation property, 44 with the Levi property, 89 with the Riesz decomposition property,

43 with the strong Levi property, 89 with vertex x, 2 with vertex at zero, 2 with weakly compact base, 123 Yudin, 125

cone base, 39 cone with vertex x, 2 cone with vertex at zero, 2 consistent topology, 73, 248 continuity and positivity, 82 continuity of linear functionals, 84 continuity of positive operators, 82, 83 continuous pull-back cone, 152 contraction operator, 171 convergence

pointwise, 246 uniform, 105

convex combination, 254 convex cone, 2 convex function, 25 convex hull of a set, 254

closed, 254 relatively compact, 254

convex set, 2, 245 copy of a space into another space, 8 countable sup property, 34 countably additive set function, 21

decomposition of a polyhedron, 143 decomposition property, 43, 216

C-, 188 on a subcone, 188

decreasing net, 7 Dedekind cr-complete ordered vector space,

17, 109 Dedekind complete /C-lattice, 178 Dedekind complete ordered vector space,

10, 17 directed downward set, 7 directed upward set, 7 directional wedge of a set, 123 directions of a wedge, 123

Index 273

discrete vector, 37 disjoint vectors in a Riesz space, 21 dominance cone, 152 dominance ordering of an operator, 152 domination by a vector, 3 double dual, 253 double dual wedge of a cone, 70 double orthogonal complement, 149 dual

algebraic, 31, 205, 246 double, 253 norm, 253 order, 33 regular, 33 super topological, 202 topological, 63, 246

dual pair, 247 dual system, 247 dual wedge, 70

Eberlein-Smulian theorem, 255 eigenvalue, 163

leading, 166 of Krein operator, 163 of operator, 163

eigenvector, 163 of Krein operator, 163 of operator, 163

equivalent norms, 102 Euclidean norm, 256 Euclidean topology, 256 exact Riesz-Kantorovich functional, 204,

211 exact Riesz-Kantorovich transform, 185 exposed extreme point, 31 exposing linear functional, 31 extension

of £b(L,7V), 192 of Z/, 199, 202 of additive mapping, 24 of operator, 24, 26-28 of positive functionals, 33, 34 smallest, 30

extension cone ,191 extremal ray, 138 extremal ray of cone, 37, 40 extremal vector of cone, 37 extreme point

exposed, 31 of polyhedron, 133 strongly exposed, 102

extreme point of a convex set, 36 extreme points of polyhedra, 133

4>j, functions vanishing off finite subsets of J, 129

[£], full hull of set £ , 6

/ > 0, positive linear functional, 31 face of convex set, 36 Farkas alternative, 149 filter base, 244 finite dimensional separating hyperplane

theorem, 252 finite-rank operator, 161 fixed point of a function, 98 Frechet lattice, 207 full hull of a set, 6 full set, 5 function

additive, 23, 184 affine, 221, 227 concave, 182 convex, 25 homogeneous, 23 linear, 227 lower order bounded, 184 monotone, 184 open, 257 order bounded, 184 piecewise affine, 222, 228 piecewise linear, 222, 228 positively homogeneous, 25, 184 rational, 47 strictly positive, 53 subadditive, 25, 184 sublinear, 25, 184 super additive, 184 superlinear, 184 upper order bounded, 184 upper semicontinuous, 199

functional exact Riesz-Kantorovich, 204 extendable, 33, 34 generalized Riesz-Kantorovich, 205 Riesz-Kantorovich, 204 supporting a set, 31

gauge of a set, 246 generalized ice cream cone, 102 generalized Riesz-Kantorovich functional,

205 generating cone, 4 graph of function, 259 greatest lower bound of a set, 6 Grothendieck's theorem, 255

Hahn-Banach extension theorem, 26, 253 half-ray, 37, 40 Hamel basis, 124 Hilbert space adjoint operator, 109 homogeneous function, 23 homogeneous linear inequality, 131 hyperinvariant subspace, 165 hyperplane, 232, 234

274 Index

oriented, 234 hyperplane orientation, 234

L x , ideal generated by x, 103 inf A, the infimum of the set A, 6 A A, the /C-infimum of set A, 175 ice cream cone, 41, 99

generalized, 102 ideal, 27, 217

in a vector lattice, 18 principal, 217

ideal generated by a vector, 103 increasing net, 7 inequality

BirkhofFs, 21 linear, 131 triangle, 21, 245, 253

infimum, 175 /C-, 175 of a set, 6 of functionals, 56 of operators, 54 of two vectors, 6

infinite distributive laws, 20 infinite interpolation property, 54 integral operator, 161 interior point of a cone, 64 interior separating hyperplane theorem, 252 intermediate vector between two sets, 44 internal point of a set, 5, 251 interpolation property, 44

infinite, 54 strong, 54

invariant subspace, 165 isomorphic topological vector spaces, 244 isomorphism notion, xiii

James's theorem, 255

/C-infimum, 175 AC-lattice, 177 /C-lattice cone, 177 /C-sublattice, 181

generated by A, 181 /C-supremum, 174 kernel of integral operator, 161 Krein operator, 160 Krein space, 107, 159, 160 Krein-Smulian theorem, 254

L x , ideal generated by x, 103 C\y{L, M) , the vector space of order

bounded operators from L to M, 23 C(X, Y), linear operators from X to Y, 23 CT(L, M) , the vector space of regular

operators from L to M, 23 £^_(L, iV), extension cone, 191

Z/, the topological dual of L, 246 L'+, the wedge of all positive continuous

linear functionals, 63 ^i-type sequence, 109 ^oo(5'), the bounded real functions on 5,

170 L~, the order dual of L, 33 L r , the regular dual of L, 33 lattice cone, 13, 14 lattice copy of a Riesz space, 19 lattice embeddable Riesz space, 19 lattice homomorphism, 19 lattice identities, 14 lattice inequalities, 21 lattice isomorphic Riesz spaces, 19 lattice isomorphism, 19 lattice norm, 85 lattice ordering, 13 lattice-subspace, 19 leading eigenvalue, 166 least upper bound of a set, 6 Levi property, 89

strong, 89 lexicographic cone, 11 lexicographic ordering, 11 lexicographic plane, 11 linear

additive, 23 linear function, 23, 227 linear functional, 31

exposing a point of a set, 31 order bounded, 31 positive, 31 regular, 31 strictly positive, 31 strongly exposing a point, 102

linear inequality, 131 homogeneous, 131

linear programming principle, 144 linear topology, 244

bornological, 111 linear transformation, 8 linearly independent set, 124 locally convex space, 245

bornological, 111 locally convex topology, 245

compatible, 248 consistent, 248 generated by a family of seminorms, 245

lower bound of a set, 6 lower order bounded function, 184 lower order bounded mapping, 184

Mackey topology, 251, 254 Mackey's theorem, 248 Mackey-Arens theorem, 250 majorizing set, 3

Index 275

majorizing subspace, 28 majorizing vector subspace, 9 mapping

additive, 23, 184 bilinear, 247 lower order bounded, 184 monotone, 56, 78, 184 open, 257 order bounded, 184 positively homogeneous, 184 sign, 234 subadditive, 184 sublinear, 25, 184 superadditive, 184 superlinear, 184 upper order bounded, 184

Mazur's theorem, 254 measure, 22

signed, 21 metric induced by a norm, 253 metrizability

of unit ball, 254 weak, 254 weak*, 254

metrizable topological vector, 245 minimal set of inequalities, 139 Minkowski functional of a set, 246 modulus of operator, 57 monotone function, 184 monotone mapping, 56, 78, 184 monotone net, 7 monotone norm, 86

N, the natural numbers, { 1 , 2 , . . . } , 243 r-normal cone, 76 x~~, negative part of vector cc, 14 negative part

of functional, 54, 56 of vector, 14

net Cauchy, 245 decreasing, 7 increasing, 7 monotone, 7

nontrivial vector subspace, 165 norm, 252

equivalent to another norm, 102 Euclidean, 256 lattice, 85 monotone, 86

norm dual, 253 norm topology, 253 normal cone, 76, 106, 108 normal to a hyperplane, 234 normed Riesz space, 85 normed space, 253

ordered, 85

normed vector lattice, 85 null space of operator, 149

£°(L, AT), order extension of £ b (L, iV) , 192 x _L y, orthogonal vectors, 21 one-sided bipolar of a set, 135, 249 one-sided bipolar theorem, 135, 249 one-sided polar, 134 open function, 257 open mapping, 257 open mapping theorem, 257 operator, 8, 23, 257

adjoint, 148 compact, 161 composition, 170 contraction, 171 finite-rank, 161 integral, 161 Krein, 160 order bounded, 23 order-embedding, 8 positive, 23 quasinilpotent, 166 rank-one, 161 regular, 23 strictly positive, 23

operator alternative, 148 order bounded function, 184 order bounded linear functional, 31 order bounded mapping, 184 order bounded operator, 23 order bounded subset, 6 order complete ordered vector space, 10, 17 order dual of an ordered vector space, 33 order extension

of £ b (L, iV) , 192 of ! / , 199

order in a vector space, 3 order interval, 5 order isomorphic ordered vector spaces, 8 order isomorphism, 8 order relation, 3 order summable sequence, 109 order topology, 110

generated by a cone, 110 order unit for a wedge, 5 order-convex set, 5 order-embeddable ordered vector space, 8 order-embedding, 8

topological, 156 ordered Banach space, 85

with the Levi property, 89 with the strong Levi property, 89

ordered normed space, 85 ordered topological vector space, 62 ordered vector space, 3

cr-order complete, 17, 109

276 Index

almost Archimedean, 12 Archimedean, 11 Dedekind cr-complete, 17, 109 Dedekind complete, 10, 17 order complete, 10, 17 order embeddable, 8 Riesz space, 13 uniformly complete, 105 with the countable sup property, 34

orientation of a hyperplane, 234 oriented arrangement of hyperplanes, 234 oriented hyperplane, 234 orthogonal complement of a set, 149 orthogonal vectors in a Riesz space, 21

A , # , one-sided bipolar of set A, 135, 249 A*, one-sided polar of set A, 249 L+, L+, the positive cone of the ordered

vector space L, 3 T > 0, positive operator, 23 A°, polar of set A, 248 A°°, bipolar of set A, 249 x + , positive part of vector x, 14 partially ordered set, 3 partially ordered vector space, 3 piecewise afrme function, 222, 228 piecewise linear function, 222, 228 point

extreme of a convex set, 36 fixed of a function, 98 interior of a cone, 64 internal, 5, 251 quasi-interior, 108 strongly exposed, 102

point separation by L~, 217 pointed convex cone, 2 pointed convex cone with vertex at zero, 2 pointwise convergence, 247 polar

absolute, 248 one-sided, 134, 249 two-sided, 135, 248

polar of a set, 135, 248 polyhedral cone, 131 polyhedral wedge, 131

generated by functionals, 131 polyhedron, 131 polyhedron decomposition, 143 polytope, 134 positive basis, 125 positive cone of an ordered vector space, 3 positive extension of operator, 24, 28 positive linear functional, 31 positive operator, 23

in Hilbert space, 109 positive part

of functional, 56

of operator, 54 of vector, 14

positive vector, 3 positively homogeneous function, 25, 184 positively homogeneous mapping, 184 pre-order in a vector space, 3 pre-order relation, 3 pre-partially ordered set, 3 primitive matrix, 161 principal ideal, 103, 217 principle

of linear programming, 144 of uniform boundedness, 254

product cone, 8 property

antisymmetry, 3 Archimedean, 11 decomposition, 43, 216 interpolation, 44 reflexivity, 3 Riesz decomposition, 43, 216 transitivity, 3

pseudo-base of a cone, 200 pull-back cone, 152, 158

continuous, 152

Q, the rational numbers, 243 quasi-interior point, 108 quasinilpotent operator, 166

R, the set of real numbers, 243 7^(A), Riesz subspace generated by A, 18 r (T) , spectral radius of operator T, 166 7Zf, Riesz-Kantorovich transform, 185 7 £ T , exact Riesz-Kantorovich transform,

185 R—K, Riesz-Kantorovich transform, 185 range of operator, 149 rank-one operator, 161 rational function, 47 reflexive Banach space, 254 reflexivity, 3 regions of piecewise afrme function, 228 regular dual of an ordered vector space, 33 regular linear functional, 31 regular operator, 23 representation of piecewise afrine function,

222 Riesz decomposition property, 43, 216 Riesz homomorphism, 19 Riesz isomorphic Riesz spaces, 19 Riesz isomorphism, 19 Riesz space, 13

normed, 85 Riesz subspace, 18, 20, 27 Riesz subspace generated by a set, 18 Riesz-Kantorovich formulas, 57

Index 277

Riesz-Kantorovich functional, 204 exact at a vector, 211 generalized, 205

Riesz-Kantorovich theorem, 54 Riesz-Kantorovich transform, 185, 204

T ^> 0, strictly positive operator, 23 Y.A, the /C-supremum of set A, 175 6-topology, 250 O'(L), super topological dual of L, 202 <7-additive set function, 21 cr-order complete cone, 109 (j-order complete ordered vector space, 17,

109 sup A, the supremum of the set A, 6 / ^> 0, strictly positive linear functional, 31 r(T), spectral radius of operator T, 166 seminorm, 245 separable topological space, 76 separating hyperplane theorem

algebraic, 251 finite dimensional, 252 interior, 252 strong, 252

separation of points, 217 from closed convex sets, 252

separation of sets, 251 strong, 252

sequence of ix-type, 109 order summable, 109 uniformly Cauchy, 105 uniformly convergent, 105

set absorbing, 33, 110, 244 absorbing another set, 110 balanced, 6, 244 bipolar, 249 bounded above, 6 bounded below, 6 circled, 6, 244 compact, 255 complete, 245 comprehensive, 32 convex, 2, 245 directed downward, 7 directed upward, 7 full, 5 linearly independent, 124 majorizing, 3 one-sided bipolar, 135, 249 order bounded, 6 order interval, 5 order-convex, 5 partially ordered, 3 polar, 248 pre-partially ordered, 3

solid, 27, 98 supported by a functional, 31 topologically bounded, 77, 246 weakly compact, 255

sign mapping, 234 signed measure, 21

with finite total variation, 22 smallest extension, 30 solid domain, 228 solid set, 27, 98 space

Krein, 107, 159, 160 null of operator, 149

spectral radius of operator, 166 Stiemke alternative, 149 Stone-Weierstrass approximation theorem,

223 strict domination by a vector, 3 strictly positive function, 53 strictly positive linear functional, 31 strictly positive operator, 23 strictly positive vector, 73 strong interpolation property, 54 strong Levi property, 89 strong separating hyperplane theorem, 252 strong separation of sets, 252 strong topology, 251 strongly exposed extreme point, 102 strongly exposing linear functional at a

point, 102 subadditive function, 25, 184 subadditive mapping, 184 subcone, 37 sublinear function, 184 sublinear mapping, 25, 184 subspace

/C-sublattice, 181 lattice-subspace, 19 majorizing, 28 Riesz, 18

super cone, 174 super topological dual, 202 super topological dual cone, 199 superadditive function, 184 superadditive mapping, 184 superlinear function, 184 superlinear mappping, 184 supporting functional, 31 supremum, 174

/C-, 174 of a set, 6 of functionals, 56 of operators, 54 of two vectors, 6

T > 0, positive operator, 23 T ^> 0, strictly positive operator, 23

278 Index

X', topological dual of X, 63 r-complete topological vector space, 245 r-normal cone, 76 theorem

algebraic separating hyperplane, 251 Banach-Steinhaus, 254 bipolar, 249 closed graph, 259 Eberlein-Smulian, 255 finite dimensional separating hyperplane,

252 Grothendieck, 255 Hahn-Banach , 253 interior separating hyperplane, 252 James, 255 Krein-Smulian, 254 Mackey, 248 Mackey-Arens, 250 Mazur, 254 one-sided bipolar, 249 open mapping, 257 Riesz-Kantorovich, 54 strong separating hyperplane, 252

topological dual, 63, 246 topological order-embeddability, 156 topological vector space, 244

metrizable, 245 ordered, 62 topologically complete, 245

topologically bounded set, 77, 246 topology

G-, 250 compatible, 73 consistent, 73 Euclidean, 256 linear, 244 locally convex, 245 Mackey, 251, 254 norm, 253 of pointwise convergence, 246, 247 of uniform convergence, 250 order, 110 strong, 251 weak, 246, 247, 253 weak*, 253

total ordering, 11 total variation of a signed measure, 22 transformer, 31 transitivity, 3 triangle inequality, 245, 253 triangle inequality in Riesz spaces, 21 two-sided polar of a set, 248

U, the closed unit ball of L, 253 £/', the closed unit ball of V', 253 £/", the closed unit ball of L"', 253

Xn -^x(v), sequence {xn} f-uniformly convergent to x, 105

uniform boundedness principle, 254 uniform convergence, 105 uniformly Cauchy sequence, 105 uniformly complete ordered vector space,

105 uniformly convergent sequence, 105 unit ball, 253

metrizable, 254 upper bound of a set, 6 upper order bounded function, 184 upper order bounded mapping, 184 upper semicontinuous function, 199

vector discrete, 37 extremal of cone, 37, 138 intermediate, 44 positive, 3 strictly positive, 73

vector lattice, 13 normed, 85

vector ordering, 3 induced on a vector subspace, 7 lattice, 13 lexicographic, 11 total, 11

vector pre-ordering, 3 vector space

normed, 253 ordered, 3 partially ordered, 3 topological, 244

vector sublattice, 18, 27 vector sublattice generated by a set, 18 vector subspace

cofinal, 9 majorizing, 9

vectors disjoint in a Riesz space, 21 orthogonal in a Riesz space, 21

W(5) , wedge generated by the set 5, 122 u>, weak topology of L, 253 w*, weak* topology of I / , 253 weak compactness, 254, 255 weak topology, 246, 247, 253 weak* topology, 253 weakly compact set, 255 wedge, 2

directional, 123 double of a wedge, 70 dual of a cone, 70 dual of a given wedge, 70 generated by a set, 122 polyhedral, 131

Index 279

wedge alternative, 148

X', topological dual of X, 63 X*, algebraic dual of X} 31 x+, positive part of vector x, 14 x~, negative part of vector x, 14

Yudin basis, 125 Yudin cone, 125

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