Functional analysis with applications

406 Pages·2019·1.538 MB·English

by  Georgiev S.G., Zennir K

SvetlinG.Georgiev,KhaledZennir
FunctionalAnalysiswithApplications
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Svetlin G. Georgiev, Khaled Zennir
Functional
Analysis with
Applications
|
MathematicsSubjectClassification2010
46-00,46-01,46B25,46E15,46E20,46E30
Authors
Prof.Dr.SvetlinG.Georgiev
KlimentOhridskiUniversityofSofia
DepartmentofDifferentialEquations
FacultyofMathematicsandInformatics
1126Sofia
Bulgaria
[email protected]
Dr.KhaledZennir
QassimUniversity
DepartmentofMathematics
BuraydahAl-Qassim51452
SaudiArabia
[email protected]
ISBN978-3-11-065769-2
e-ISBN(PDF)978-3-11-065772-2
e-ISBN(EPUB)978-3-11-065804-0
LibraryofCongressControlNumber:2019937688
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
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Typesetting:VTeXUAB,Lithuania
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Preface
Functionalanalysismeansanalysisonfunctionspaces.Thisisafieldofmathematics
thatdevelopedinthefirsthalfofthe20thcenturythanksinparticulartotheworkof
M.Frechet,S.Banach,andD.Hilbert.Examplesoftheefficiencyoffunctionalanalysis
hasbeentheintroductionofSobolev’sspaces(1935)andL.Schwartz’sinventionofthe
theoryofdistributions(1945–1950).Thesespaceshavemadegreatprogressinsolving
the problems of partial differential equations and provide the main tools still used
todayinthisfieldofboththeoreticalandnumericalstudies.
Classicalanalysisfocusesonfinitedimensionalspacesonℝorℂ.Thisissuitable,
forexample,forsolvinglineardifferentialequations.Inordertosolvemorecompli-
catedequations,likenonlineardifferentialequations,integralequations,andpartial
differentialequations,thesolutionshavetobesoughtaprioriinvectorspacesofan
infinitenumberofdimensions.Thecomputationofexplicitsolutionsisoftenoutof
reachandonetriestodescribethestructureofthesesolutionsbytheirbelongingto
spacesadaptedtotheproblemposed.Thestudyofstabilitynaturallyleadstocon-
sideringspaceswithtopologiesdefinedbynorms,semi-normsordistances.Froma
purelymathematicalpointofview,functionalanalysiscanalsobeseenasanexten-
siontoinfinitedimensionsofEuclideangeometryinfinitedimensions.Thetransition
fromfinitedimensiontoinfinitedimensionisnotalwayseasybecauseweloseapart
ofthegeometricintuition.Whereasonafinitedimensionalvectorspacethereisonly
one“reasonable”topology,onaspaceofinfinitedimensionwemustoftenconsider
severaltopologiessimultaneously.
Themaingoal,inrealizingthistextbook,istopresentausefultooltojuniorre-
searchersandbeginninggraduatestudentsofengineeringandsciencecoursesinor-
dertoacquireelementaryknowledgeandsolidtoolsthatarefundamentaltotheun-
derstandingofmathematicsandtheparticulardisciplines(geometry,probabilities,
partialdifferentialequations)withinphysics,inmechanics,orintheapplicationsof
mathematicstotheanalysisoflargesystems.Itcontainstenchapters,andeachchap-
terconsistsofresultswiththeirdetailedproofs,numerousexamples,andexercises
withsolutions.Eachchapterconcludeswithasectionfeaturingadvancedpractical
problemswithsolutionsfollowedbyasectiononnotesandreferences,explainingits
contextwithinexistingliterature.Wewillpresenthereinadetailedwaythecontents
ofeachofthem.
Chapter1isentirelydevotedtothepresentationofdefinitionsandresultsneces-
saryforproceedinginthiswork.Wefirstrecallafewbasicresultsonthelinear,metric,
normedandBanachspacesanditsproperties.Theseareusedinparticulartointro-
ducethevariousconceptsofweaksolutionstoPDEs.Wewillseeregularlylinksand
relationshipsbetweenfunctionanalysisandapplicationsonPDEs.Chapter2istitled
Lebesgueintegration.Itisdevotedtothestudyofmeasureandintegration,Lebesgue
measurablefunctionsandgeneralmeasurespaces,wheretherearemanyprovedre-
https://doi.org/10.1515/9783110657722-201
VI | Preface
sults.ThepurposeofChapter3istopresentresultsaccordingtotheLpspaces,which
contains,definitions,separability,dualityandgeneralLpspaceswithitsnorms.The
results,presentedinChapter4,concernlinearoperators,inverseoperatorsinnormed
linearspacesandtheirproperties.Chapter5istitledLinearfunctionals;herewein-
troduceandtreatthelinearfunctionalsintheirgeneralformandrelatedtheadjoint
operators.Chapter6isreservedfortopologicalstudies;itisfollowedbyChapter7ti-
tledSelf-adjointoperatorsinHilbertspaces.Themethodofthesmallparameterwill
bethemainsubjectofChapter8andtheparametercontinuationmethodwillbethe
subjectofChapter9.
So we realize that the fixed-point theorems are essential in the applications of
the function analysis. They are the basic mathematical tools in showing the exis-
tence of solutions in various kinds of equations. Fixed-point theory is at the heart
ofnonlinearanalysisandprovidesthenecessarytoolstostudyexistencetheorems
in many different nonlinear problems. The aim of Chapter 10 is the study of some
fixed-pointtheorems.Westartwiththesimplestandbestknownofthem:Banach’s
fixed-pointtheoremforcontractionmaps.ThenweaddresstheBrinciarifixed-point
theorem,whichisageneralizationofthistheorem.Wewillthenseemorepowerful
andsomewhatdeepertheorems.Wecanthusstudysuccessivelythetheoremofthe
fixedpointofBrouwer(validinfinitedimension)andthenthetheoremofthefixed
pointofSchauder(whichisthegeneralizationininfinitedimension).UnlikeBanach’s
theorem,theproofsofthelattertworesultsarenotconstructive,whichexplainswhy
they require somewhat more sophisticated tools. Many different proofs of these re-
sultsexistandonemaybeinterestedinoneormoreofthem.Wefinishthischapter
bygivingapplicationsinmanyproblems.
Thisisthefirstvolumeofaseriesofatleasttwovolumes;theremainderofthe
serieswillbepreparedlater.
SvetlinG.Georgiev
KhaledZennir
Contents
Preface|V
1 Vector,metric,normedandBanachspaces|1
1.1 Vectorspaces|1
1.2 Metricspaces|24
1.3 Usefulinequalities|29
1.4 Completespaces|32
1.5 Normedspaces|39
1.6 Banachspaces|53
1.7 Innerproductspaces|54
1.8 Hilbertspaces|62
1.9 Separablespaces|69
1.10 Advancedpracticalproblems|70
2 Lebesgueintegration|73
2.1 Lebesgueoutermeasure.Measurablesets|73
2.2 TheLebesguemeasure.TheBorel–Cantellilemma|95
2.3 Nonmeasurablesets|99
2.4 TheCantorset.TheCantor–Lebesguefunction|102
2.5 Lebesguemeasurablefunctions|107
2.6 TheRiemannintegral|124
2.7 Lebesgueintegration|125
2.7.1 TheLebesgueintegralofaboundedmeasurablefunctionoverasetof
finitemeasure|125
2.7.2 TheLebesgueintegralofameasurablenonnegativefunction|135
2.7.3 ThegeneralLebesgueintegral|143
2.8 Continuityanddifferentiabilityofmonotonefunctions.Lebesgue’s
theorem|158
2.9 Generalmeasurespaces|168
2.10 Generalmeasurablefunctions|169
2.11 Integrationovergeneralmeasurespaces|171
2.12 Advancedpracticalproblems|176
3 TheLpspaces|179
3.1 Definition|179
3.2 TheinequalitiesofHölderandMinkowski|180
3.3 Someproperties|182
3.4 TheRiesz–Fischertheorem|183
3.5 Separability|189
VIII | Contents
3.6 Duality|190
3.7 GeneralLpspaces|204
3.8 Advancedpracticalproblems|207
4 Linearoperators|209
4.1 Definition|209
4.2 Linearoperatorsinnormedvectorspaces|211
4.3 Inverseoperators|229
4.4 Advancedpracticalproblems|233
5 Linearfunctionals|235
5.1 TheHahn–Banachextensiontheorem|235
5.2 ThegeneralformofthelinearfunctionalsonEninthecaseF=R|241
5.3 ThegeneralformofthelinearfunctionalsonHilbertspaces|242
5.4 Weakconvergenceofsequencesoffunctionals|244
5.5 Advancedpracticalproblems|244
6 Relativelycompactsetsinmetricandnormedspaces.Compact
operators|247
6.1 Definitions.Generaltheorems|247
6.2 Criteriaforcompactnessofsetsinmetricspaces|250
6.3 ACriteriaforrelativecompactnessinthespaceC([a,b])|254
6.4 ACriteriaforcompactnessinthespaceLp([a,b]),p>1|257
6.5 Compactoperators|260
6.6 Advancedpracticalproblems|263
7 Self-adjointoperatorsinHilbertspaces|265
7.1 Adjointoperators.Self-adjointoperators|265
7.2 Unitaryoperators|266
7.3 Projectionoperators|267
8 Themethodofthesmallparameter|273
8.1 Abstractfunctionsofarealvariable|273
8.2 Powerseries|280
8.3 AnalyticabstractfunctionsandTaylor’sseries|282
8.4 Themethodofthesmallerparameter|286
8.5 Anapplicationtointegralequations|289
9 Theparametercontinuationmethod|299
9.1 Statementofthebasicresult|299
9.2 Anapplicationtoaboundaryvalueproblemforaclassofsecondorder
ordinarydifferentialequations|300