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1、LectureNotesNonlinearFunctionalAnalysis2Chapter1Lesson11.1WhatcanandcannotbedonewithLi-nearFunctionalAnalysisInlastsemesterscourse,wehavestudiedvariouspropertiesofBanachSpacesandHilbertSpacesandbrie
ydiscussedanapplicationtolineardi erentialequationsviaSobolevspacesinparticular.Wew
2、illnowviewthattopicincloserdetail.Theorem1.1.1.(Rieszrepresentationtheorem)LetHbeaHilbertspace,withaninnerproducth;i.LetL:H!Rbealinearboundedoperator.Thenthereexistsauniquey2HsuchthatL(x)=hx;yiforallx2HLemma1.1.2.(Cauchy-Schwarzinequality)LetHbeaHilbertspacewithinnerproducth;ia
3、ndnormkk.Thenforallx,y2Hjhx;yijkxkkykLetn2NandRn.De nition1.1.3.L2()isthespaceofallfunctionsf:!Rsuchthatthenorm012Zkfk:=@f(x)2dxA234Lesson1is nite.Letf,g2L2().L2()isaHilbertspacewithrespecttotheinnerproductZhf;gi2:=f(x)g(x)dxDe nition1.1.4.LetC1()bethespaceofallfunctionsf:!Rthat
4、0arein nitelydi erentiableandforwhichf(n)(x)=0forx2@andn2N.Letf,g2C1andde neaninnerproductonC1through00Zhf;gi0;1=(f(x)g(x)+rf(x)rg(x))dxThentheSobolevspaceH1()isthecompletionofthespaceC1()with001respecttothenormkk:=h;i20;10;1Question:Whydowede netheSobolevspaceH1()?0Answer:Wewi
5、llbeginansweringthisquestionbylookingatthefollowingexample:Example1.1.5.ConsiderthefollowingDirichletproblem:Findafunctionu2C2()C()suchthat(
