Finite elements in vector lattices

230 Pages·2014·1.327 MB·English

by  Martin R. Weber

MartinR.Weber
FiniteElementsinVectorLattices
Alsoof Interest
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ISBN978-3-11-026303-9,e-ISBN978-3-11-026334-3
AbstractAlgebra.ApplicationstoGaloisTheory,AlgebraicGeometry
andCryptography
Carstensen,Fine,Rosenberger,2011
ISBN978-3-11-025008-4,e-ISBN978-3-11-025009-1
JournalofNumericalMathematics
Hoppe,Kuznetsov(Eds.-in-Chief)
ISSN1570-2820,e-ISSN1569-3953
Martin R. Weber
Finite Elements in
Vector Lattices
|
MathematicsSubjectClassiication2010
46B40,46B42,46A40,46E05,47B65,06F25
Author
Prof.Dr.MartinR.Weber(Seniorprofessor)
TUDresden
InstitutfürAnalysis
Helmholtzstr10
01062Dresden
[email protected]
ISBN978-3-11-035077-7
e-ISBN978-3-11-035078-4
Set-ISBN978-3-11-035079-1
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ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress.
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
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detailedbibliographicdataareavailableintheInternetathttp://dnb.dnb.de.
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Typesetting:PTP-Berlin,Protago-TEX-ProductionGmbH,Berlin
Printingandbinding:CPIbooksGmbH,Leck
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|
Gewidmet
meinerliebenFrauUte,
meinenliebenKindern
AnnettundAlexander
und
meinemverehrtenLehrer
BorisMichailowitschMakarow

Contents
1 Introduction|1
2 Orderedvectorspacesandvectorlattices|4
2.1 Orderedvectorspacesandpositiveoperators|4
2.2 Vectorlattices|6
2.3 Orderednormedspaces|11
2.4 NormedRieszspacesandBanachlattices|12
2.5 RepresentationofBanachlattices|16
3 Finite,totallyiniteandselfmajorizingelements|18
3.1 Finiteandtotallyiniteelementsinvectorlattices|18
3.2 FiniteelementsinBanachlattices|29
3.3 FiniteelementsinsublatticesandindirectsumsofBanach
lattices|33
3.3.1 Finiteelementsinsublattices|33
3.3.2 FiniteelementsinthebidualofBanachlattices|37
3.3.3 FiniteelementsindirectsumsofBanachlattices|39
3.4 Selfmajorizingelementsinvectorlattices|41
3.4.1 Theorderidealofallselfmajorizingelementsinavectorlattice|42
3.4.2 Generalpropertiesofselfmajorizingelements|44
3.4.3 Examplesofselfmajorizingelements|47
3.5 Finiteelementsinℓ-algebrasandinproductalgebras|49
3.5.1 Latticeorderedalgebras|49
3.5.2 Finiteelementsinunitaryℓ-algebras|52
3.5.3 Finiteelementsinnonunitary-algebras|57
3.5.4 Finiteelementsinproductalgebras|63
4 Finiteelementsinvectorlatticesoflinearoperators|69
4.1 Somegeneralresults|70
4.2 Finitenessofregularoperatorson-spaces|75
4.3 Finiterankoperatorsinthevectorlatticeofregularoperators|77
4.4 SomevectorlatticesandBanachlatticesofoperators|81
4.4.1 Vectorlatticesofoperators|83
4.4.2 Banachlatticesofoperators|84
4.5 Operatorsasiniteelements|90
4.6 Finiterankoperatorsasiniteelements|92
4.7 ImpactoftheorderstructureofV(,)onthelatticepropertiesof
and|96
viii | Contents
5 Thespaceofmaximalidealsofavectorlattice|100
5.1 Representationofvectorlatticesbymeansofextendedrealcontinuous
functions|100
5.2 Maximalidealsanddiscretefunctionals|103
5.3 Thetopologyonthespaceofmaximalidealsofavectorlattice|107
5.4 TheHausdorfpropertyof M|109
6 Topologicalcharacterizationofiniteelements|115
6.1 Topologicalcharacterizationofinite,totallyiniteandselfmajorizing
elements|115
6.1.1 Thecanonicalmapandtheconditionalrepresentation|116
6.1.2 Topologicalcharacterizationofiniteelements|121
6.1.3 Topologicalcharacterizationoftotallyiniteelements|125
6.1.4 Topologicalcharacterizationofselfmajorizingelements|129
6.2 Relationsbetweentheidealsofinite,totallyiniteandselfmajorizing
elements|131
6.3 ThetopologicalspaceMforvectorlatticesoftype(Σ)|134
6.4 Examples|138
7 Representationsofvectorlatticesandtheirproperties|144
7.1 Aclassiicationofrepresentationsandthestandardmap|144
7.2 Vectorlatticesoftype(Σ)andtheirrepresentations|148
8 Vectorlatticesofcontinuousfunctionswithiniteelements|157
8.1 Vectorlatticesofcontinuousfunctionswithmanyinite
functions|157
8.2 Finiteelementsinvectorlatticesofcontinuousfunctions|162
8.3 Anisomorphismresultforvectorlatticesofcontinuous
functions|167
9 Representationsofvectorlatticesbymeansofcontinuous
functions|171
9.1 Representationswhichcontaininitefunctions|171
9.2 Theexistenceof Φ-representationsforvectorlatticesof
type(Σ)|177
9.3 -vectorlattices|182
9.4 Vectorlatticesoftype( )|184

10 Representationsofvectorlatticesbymeansofbasesofinite
elements|191
10.1 Basesofiniteelementsand-representations|191
Contents | ix
10.2 RepresentationsbymeansofR-basesofiniteelements|195
10.3 Somepropertiesoftherealizationspace|199
ListofExamples|207
ListofSymbols|209
Bibliography|211
Index|217