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1Introduction 1

2Ordered vector spaces and vector lattices 4

2.1Orderedvector spaces and positiveoperators 4

2.2Vector lattices 6

2.3Ordered normed spaces 11

2.4Normed Riesz spaces and Banach lattices 12

2.5Representation of Banach lattices 16

Finite, totally finite and selfmajorizing elements 18

3.1Finite and totally finite elements in vector lattices 18

3.2Finite elements in Banach lattices 29

3.3Finite elements in sublattices and in direct sums of Banach lattices 33

      3.3.1Finite elements in sublattices 33

      3.3.2Finite elements in the bidualof Banach lattices 37

      3.3.3Finite elements in direct sums of Banach lattices 39

3.4Selfmajorizing elements in vector lattices 41

      3.4.1The orderidealof all selfmajorizingelementsin a vectorlattice 42

      3.4.2Generalproperties of selfmajorizingelements 44

      3.4.3Examples of selfmajorizing elements 47

3.5Finite elementsine-algebras and in product algebras 49

      3.5.1Lattice ordered algebras 49

      3.5.2Finite elements in unitary e-algebras 52

      3.5.3Finite elements in nonunitaryf-algebras 57

      3.5.4Finite elements in product algebras 63

4Finite elements in vector lattices of linear operators 69

4.1Some general results 70

4.2Finiteness of regular operators on AL-spaces 75

4.3Finite rankoperators in the vector lattice of regularoperators 77

4.4Some vector lattices and Banach lattices of operators 81

      4.4.1Vector lattices of operators 83

      4.4.2Banach lattices of operators 84

4.5Operators as finite elements 90

4.6Finite rank operators as finite elements 92

4.7Impact of the order structure of V(E,F)on the lattice properties of E and F 96

5The space of maximalideals of a vector lattice 100

5.1Representation ofvectorlattices by means of extended real continuous functions 100

5.2Maximalideals and discrete functionals 103

5.3The topology on the space of maximalideals of avector lattice 107

5.4The Hausdorff property of M 109

Topological characterization of finite elements 115

6.1Topologicalcharacterization of finite,totally finite and selfmajorizing elements 115

      6.1.1The canonical map and the conditional representation 116

      6.1.2Topologicalcharacterization of finite elements 121

      6.1.3Topologicalcharacterization of totally finite elements 125

      6.1.4Topologicalcharacterization of selfmajorizingelements 129 6.2Relations between the ideals of finite,totally finite and selfmajorizing elements 131

      6.3The topological space Μ for vector lattices of type(Σ) 134

      6.4 Examples 138

7Representations of vector lattices and their properties 144 C2\7.1A classification of representations and the standard map 144

7.2Vector lattices of type (Σ)and their representations 148

8Vector lattices of continuous functions with finite elements 157

8.1Vector lattices of continuous functions with many finite functions 157

8.2Finite elements invectorlattices of continuous functions 162

8.3An isomorphism result forvectorlattices of continuous functions 167

9Representations of vector lattices by means of continuous Functions 171

9.1Representations which contain finite functions 171

9.2

The existence of Φα-representations for vector lattices of type (Σ) 177

9.3LF-vector lattices 182

9.4Vector lattices of type (CM) 184

10Representations of vector lattices by means of bases of finite elements 191

10.1Bases of finite elements and α-representations 191

10.2Representations by means of R-bases of finite elements 195

10.3Some properties of the realization space 199

List of Examples 207

List of Symbols 209

Bibliography 211

Index 217

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