a basis theory primer

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1、CorrectedJanuary1998RevisedJuly1997ManuscriptJuly1987ABASISTHEORYPRIMERChristopherHeilSchoolofMathematicsGeorgiaInstituteofTechnologyAtlanta,Georgia30332-0160http://www.math.gatech.edu/heiliiic1997byChristopherHeiliiiCONTENTSI.Preliminaries11.NotationandFunctionalAnalysisReview2II.Conve

2、rgenceofSeries202.UnconditionalConvergenceofSeriesinBanachSpaces213.UnconditionalConvergenceofSeriesinHilbertSpaces32III.BasesinBanachSpaces384.BasesinBanachSpaces395.AbsolutelyConvergentBasesinBanachSpaces466.SomeTypesofLinearIndependenceofSequences477.BiorthogonalSystemsinBanachSpaces5

3、08.DualityforBasesinBanachSpaces569.UnconditionalBasesinBanachSpaces5810.WeakandWeakBasesinBanachSpaces65IV.BasesandFramesinHilbertSpaces11.RieszBasesinHilbertSpaces7312.FramesinHilbertSpaces81References93ivPREFACEBasesareessentialtoolsinthestudyofBanachandHilbertspaces.Thismanuscriptpr

4、esentsaquickandhopefullyeasyintroductiontobasistheoryforreaderswithamodestbackgroundinrealandfunctionalanalysis.Ashortreviewoftherelevantbackgroundfromanalysisisincluded.Thismanuscriptgrewoutofasetofnotesoriginallypreparedin1987attheinstigationofmyPh.D.thesisadvisor,ProfessorJohnBenedett

5、ooftheUniversityofMaryland,CollegePark.Atthattime,thenow-ubiquitouseldofwaveletswasinitsinfancy.Animportantgoalofthenewtheorywastheconstructionofgood"basesorbasis-likesystemscalledframesforfunctionspacessuchasL2(R).Asolidunderstandingofbasistheorywasthereforeneeded,andthesenotesaretheo

6、springofthatneed.Theresultspresentedhereweredrawnfrommanysources,butespeciallyfromtheindispensablebooksbyLindenstraussandTzafriri[LT77],Marti[Mar69],Singer[Sin70],andYoung[You80].Asidefromafewminorresultsinconnectionwithframetheorythatareclearlyidentied,noresultspresentedinthismanuscri

7、ptareoriginalorareclaimedasoriginal.Outline.Intherstpartofthemanuscript,consistingofChapter1,wepresentareviewofbasicfunctional-analyticbackgroundmaterial.Wegivethedenitionsandthestatementsofthetheoremsthatunderliethematerialinthismanuscript,butweomittheproofs.Mostofthis