Theory_of_Functions_of_a_Real_Variable-S_Sternberg.pdf

Theoryoffunctionsofarealvariable.

ShlomoSternberg

May10,2005

Introduction.

Ihavetaughtthebeginninggraduatecourseinrealvariablesandfunctional

analysisthreetimesinthelastﬁveyears,andthisbookistheresult.The

courseassumesthatthestudenthasseenthebasicsofrealvariabletheoryand

pointsettopology.Theelementsofthetopologyofmetricsspacesarepresented

(inthenatureofarapidreview)inChapterI.

Thecourseitselfconsistsoftwoparts:1)measuretheoryandintegration,

and2)Hilbertspacetheory,especiallythespectraltheoremanditsapplications.

InChapterIIIdothebasicsofHilbertspacetheory,i.e.whatIcando

withoutmeasuretheoryortheLebesgueintegral.Theherohere(andperhaps

fortheﬁrsthalfofthecourse)istheRieszrepresentationtheorem.Included

isthespectraltheoremforcompactself-adjointoperatorsandapplicationsof

thistheoremtoellipticpartialdierentialequations.Thepdematerialfollows

closelythetreatmentbyBersandSchecterinPartialDierentialEquationsby

Bers,JohnandSchecterAMS(1964)

ChapterIIIisarapidpresentationofthebasicsabouttheFouriertransform.

ChapterIVisconcernedwithmeasuretheory.TheﬁrstpartfollowsCaratheodory’s

classicalpresentation.ThesecondpartdealingwithHausdormeasureanddi-

mension,Hutchinson’stheoremandfractalsistakeninlargepartfromthebook

byEdgar,Measuretheory,Topology,andFractalGeometrySpringer(1991).

Thisbookcontainsmanymoredetailsandbeautifulexamplesandpictures.

ChapterVisastandardtreatmentoftheLebesgueintegral.

ChaptersVI,andVIIIdealwithabstractmeasuretheoryandintegration.

ThesechaptersbasicallyfollowthetreatmentbyLoomisinhisAbstractHar-

monicAnalysis.

ChapterVIIdevelopsthetheoryofWienermeasureandBrownianmotion

followingaclassicalpaperbyEdNelsonpublishedintheJournalofMathemat-

icalPhysicsin1964.Thenwestudytheideaofageneralizedrandomprocess

asintroducedbyGelfandandVilenkin,butfromapointofviewtaughttous

byDanStroock.

Therestofthebookisdevotedtothespectraltheorem.Wepresentthree

proofsofthistheorem.Theﬁrst,whichiscurrentlythemostpopular,derives

thetheoremfromtheGelfandrepresentationtheoremforBanachalgebras.This

ispresentedinChapterIX(forboundedoperators).Inthischapterweagain

followLoomisratherclosely.

InChapterXweextendtheprooftounboundedoperators,followingLoomis

andReedandSimonMethodsofModernMathematicalPhysics.Thenwegive

Lorch’sproofofthespectraltheoremfromhisbookSpectralTheory.Thishas

theﬂavorofcomplexanalysis.ThethirdproofduetoDavies,presentedatthe

endofChapterXIIreplacescomplexanalysisbyalmostcomplexanalysis.

Theremainingchapterscanbeconsideredasgivingmorespecializedin-

formationaboutthespectraltheoremanditsapplications.ChapterXIisde-

votedtooneparametersemi-groups,andespeciallytoStone’stheoremabout

theinﬁnitesimalgeneratorofoneparametergroupsofunitarytransformations.

ChapterXIIdiscussessometheoremswhichareofimportanceinapplicationsof

thespectraltheoremtoquantummechanicsandquantumchemistry.Chapter

XIIIisabriefintroductiontotheLax-Phillipstheoryofscattering.

Contents

1Thetopologyofmetricspaces 13

1.1Metricspaces............................. 13

1.2Completenessandcompletion..................... 16

1.3NormedvectorspacesandBanachspaces.............. 17

1.4Compactness.............................. 18

1.5TotalBoundedness........................... 18

1.6Separability............................... 19

1.7SecondCountability.......................... 20

1.8ConclusionoftheproofofTheorem1.5.1.............. 20

1.9Dini’slemma.............................. 21

1.10TheLebesgueoutermeasureofanintervalisitslength. ..... 21

1.11Zorn’slemmaandtheaxiomofchoice................ 23

1.12TheBairecategorytheorem. .................... 24

1.13Tychono’stheorem.......................... 24

1.14Urysohn’slemma............................ 25

1.15TheStone-Weierstrasstheorem.................... 27

1.16Machado’stheorem. ......................... 30

1.17TheHahn-Banachtheorem...................... 32

1.18TheUniformBoundednessPrinciple................. 35

2HilbertSpacesandCompactoperators. 37

2.1Hilbertspace.............................. 37

2.1.1Scalarproducts. ....................... 37

2.1.2TheCauchy-Schwartzinequality............... 38

2.1.3Thetriangleinequality.................... 39

2.1.4Hilbertandpre-Hilbertspaces................ 40

2.1.5ThePythagoreantheorem. ................. 41

2.1.6ThetheoremofApollonius.................. 42

2.1.7ThetheoremofJordanandvonNeumann. ........ 42

2.1.8Orthogonalprojection..................... 45

2.1.9TheRieszrepresentationtheorem. ............. 47

2.1.10WhatisL 2 (T)?........................ 48

2.1.11Projectionontoadirectsum................. 49

2.1.12Projectionontoaﬁnitedimensionalsubspace........ 49

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