PDF hosted at the Radboud Repository of the Radboud University Nijmegen The following full text is a publisher's version. For additional information about this publication click this link. http://hdl.handle.net/2066/145545 Please be advised that this information was generated on 2021-10-08 and may be subject to change. seminorms on ordered vector spaces Onno van Gaans sem ι norms on ordered vector spaces een wetenschappelijke proeve op het gebied van de Natuurwetenschappen, Wiskunde en Informatica Proefschrift ter verkrijging van de graad van doctor aan de Katholieke Universiteit Nijmegen, volgens besluit van het College van Decanen in het openbaar te verdedigen op maandag 22 februari 1999 des namiddags om 3.30 uur precies door Onno Ward van Gaans geboren op 15 oktober 1971 te Waddinxveen Promotor: Prof. dr. A.C.M, van Rooij Manuscriptcommissie: Prof. dr. CD. Aliprantis, Purdue University, West Lafayette, Indiana Prof. dr. G.J.H.M. Buskes, University of Mississippi, Oxford, Mississippi Dr. B. de Pagter, Technische Universiteit Delft, Delft, The Netherlands ISBN 90-9012410-1 Contents Summary ν Samenvatting vii 1 Seminorms and Topologies on Ordered Spaces 1 1.1 Partially Ordered Vector Spaces 1 1.2 Seminorms on Ordered Spaces 3 1.2.1 Monotone, Monotone*, and Fremlin Seminorms 3 1.2.2 Monotone Seminorms and Relatively Uniform Convergence . 7 1.2.3 Additivity on the Positive Cone and Full Unit Balls 7 1.2.4 Monotone Norms and Closedness of the Positive Cone 8 1.2.5 Quotients Over Kernels of Monotone Seminorms 9 1.3 Locally Full Topologies 12 1.4 Topologies such that the Positive Cone is Closed 15 1.5 Locally Solid Topologies 16 1.6 Dual Spaces 21 2 Extending Monotone Seminorms 27 2.1 Introduction 27 2.2 Existence of Extensions 28 2.3 Uniqueness of Extensions 31 2.4 Extensions and Norm Completeness 34 3 Riesz Seminorms on Non-Riesz Spaces 37 3.1 Solid Sets in Partially Ordered Vector Spaces 37 3.2 Solvex Sets and Pre-Riesz Seminorms 42 3.3 Restricting Pre-Riesz Seminorms to Pre-Riesz Subspaces 44 3.3.1 Riesz* Homomorphisms and Pre-Riesz Spaces 44 3.3.2 Pre-Riesz Seminorms and Riesz* Homomorphisms 46 3.4 Extending Pre-Riesz Seminorms 46 3.5 Extensions and Non-Uniqueness 50 i Contents 3.6 Extensions and L- and M-Seminorms 51 3.7 Regular Seminorms: Introduction 53 3.8 Extending Regular Seminorms and Regularizations 56 3.9 Regularizations and Norm Completeness 57 3.10 Norm Completions , 59 3.10.1 Norm Completions of Normed Ordered Vector Spaces 59 3.10.2 On the Norm Completion of a Monotonely Normed Riesz Space 60 4 Subspaces of Normed Riesz Spaces 65 4.1 Introduction 65 4.2 Dual Characterization of Fremlin Seminorms 66 4.3 Embedding in a Normed Riesz Space 69 4.3.1 Construction 69 4.3.2 Preservation of Lattice Structure 71 4.3.3 No Minimality 76 4.4 On Embedding of Spaces with Unclosed Positive Cones 77 4.4.1 Embedding in a Regularly Normed Space 77 4.4.2 Embedding in a Riesz Space with a Riesz Seminorm 81 4.5 The Subspaces of an M-space 84 4.5.1 Dual Characterization of Seminorms with Full Unit Balls ... 84 4.5.2 Embedding in an M-normed Riesz Space 86 5 Spaces of Operators 89 5.1 Introduction 89 5.2 Norm Dual Spaces 91 5.2.1 Krein's Lemma and the Dual of a Regularly Normed Space . 91 5.2.2 Counterexamples 93 5.3 Bonsall's Theorem and a Riesz Space Valued Version of Krein's Lemma 95 5.4 Operators Between Normed Partially Ordered Vector Spaces 98 5.5 An Alternative for the r-Norm 101 Commentary 103 List of Symbols 107 Bibliography 109 Index 111 Curriculum Vitae 115 ii Dankwoord Velen hebben direct of indirect bijgedragen aan het totstandkomen van dit proef­ schrift en aan het plezier dat ik voortdurend in mijn werk heb gehad. Een aantal van hen wil ik met name noemen: Arnoud voor al zijn steun en hartelijkheid op elk gewenst moment; de collega's van de vakgroep voor de prettige werksfeer en hun persoonlijke belangstelling; mijn vrienden voor hun steun, belangstelling en afleid­ ing; de studenten voor het regelmatig aanwakkeren van mijn enthousiasme voor het vak; Peter en Dick, Ernie en mijnheer Bakker voor 'de muzikale omlijsting'; mijn kamergenoten Frank en Danny en Paul voor de gezellige huiskamersfeer op de werk­ plek en hun niet aflatende zorg om mijn caffeïnespiegel; and thanks to the members of the manuscript committee for carefully reading the manuscript in a very short time. Ten slotte mijn ouders en zus die mij door de dalen heen sleepten en tijdens de pieken op de grond hielden. Mijn hartelijke dank aan allen die mijn AiO-jaren tot zo'n geweldige tijd hebben gemaakt. Onno van Gaans 111 iv Summary Many of the real vector spaces occurring in analysis have a natural norm or topology as well as a natural ordering. The relation between the norm or topology and the or­ dering usually fits in the theory of normed Riesz spaces, locally full partially ordered vector spaces, or related notions. Especially for normed Riesz spaces —and Ba- nach lattices in particular— a rich theory has been developed (see e.g. Aliprantis & Burkinshaw[2], Meyer-Nieberg[17], Schaefer[22], Vulikh[23]). For normed partially ordered vector spaces, the theory includes duality (see e.g. Davies[7], Ng[19]) and spaces of operators (see e.g. Wickstead[24]). There is also theory on ordered topolog­ ical spaces (see e.g. Kelley L· Namioka[14], Namioka[18], Peressini[20], Schaefer[21] and also Aliprantis L· Burkinshaw[l] and Fremlin[8]). It is not without interest to extend theory of normed Riesz spaces to the more general setting of partially ordered vector spaces. Not merely to cover a wider range of spaces, but also because such spaces appear naturally in the theory of normed Riesz spaces itself. Indeed, linear subspaces of normed Riesz spaces are not always Riesz spaces —every partially ordered vector space is in fact a linear subspace of a Riesz space— and, also, spaces of operators between normed Riesz spaces need not be Riesz spaces. The study of normed partially ordered vector spaces is not new and neither is the study of embedding them in Riesz spaces. What seems to be new in this text is the study of seminorms on partially ordered vector spaces by viewing these spaces as subspaces of Riesz spaces. Thus, special attention has been paid to restriction, extension, and isometrical embedding problems and new results have been obtained. The properties of a seminorm ρ on a partially ordered vector space E that play important roles from many points of view are what we call the Fremlin property {—y < x < У =* p(x) < p(y))j regularity (p(x) = inf{p(r/): — y < χ < y}), and closedness of E+. Part of their significance can be found in the literature, mainly concerning spaces of operators. This text provides a more systematic treatment, dealing also with extension, isometrical embedding, norm completeness, and norm completions. Chapter 1 starts with a glossary of the basic theory of locally full and locally ν Summary solid topologies. Also, basic theory of various types of seminorms is developed, dealing with, for instance, quotients with respect to kernels and closures of positive cones. Most results are accompanied by (counter)examples. Chapter 2 studies the question whether every monotone seminorm on a par­ tially ordered vector space is the restriction of a monotone seminorm on a Riesz space. The answer is negative. A condition on a seminorm ρ that is necessary for extendability to a monotone seminorm on a Riesz space is the monotone* property (—11 < χ < v, u,v > 0 ^ p(x) < p(u) +p(v)). It is sufficient for extendability to a monotone seminorm on every partially ordered vector space, in particular every Riesz space, in which the initial space is majorizing (2.1). Fremlin seminorms on majorizing subspaces can always be extended to Fremlin seminorms on the whole space (2.3). Explicit formulas for greatest extensions are given and preservation of norm completeness is investigated (2.12). Chapter 3 studies a reasonable notion of 'Riesz seminorm' on a partially ordered vector space. The first approach is to consider seminorms that are restrictions of Riesz seminorms on the Dedekind completion. A characterization of such seminorms is given (3.15, 3.29), based on a notion of 'solid set' in a partially ordered vector space (3.7). There is, as before, an explicit construction of extensions. The ensuing notion of 'Riesz seminorm' turns out to be rather weak. For example, there may exist two such seminorms that coincide on the positive cone (3.38). A stronger notion is regularity, generalizing Riesz seminorms in a very convenient way. Theorems about norm completeness are given, generalizing results concerning the r-norm of the theory of spaces of operators (3.46). The last section studies norm completions (3.51, 3.55). Chapter 4 investigates what norms are restrictions of Riesz norms. Since every partially ordered vector space is (isomorphic to) a subspace of a Riesz space, it is very natural to ask which norms on partially ordered vector spaces allow embedding in normed Riesz spaces. The norms that can be obtained this way turn out to be Fremlin norms such that the positive cone is closed (4.8). A similar result is obtained for restriction of M-norms (4.34). Under additional conditions, the results can be extended to unclosed cones (4.21) and to seminorms (4.28).