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MIT OpenCourseWarehttp://ocw.mit.edu

18.102 Introduction to Functional AnalysisSpring 2009

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4 LECTURE NOTES FOR 18.102, SPRING 2009

Lecture1. Tuesday,3Feb.Linear spaces, metric spaces, normed spaces. Banach spaces. Examples Eu

clideanspaces,continuousfunctionsonaclosedintervalC0([0,1])withsupremumnorm. The(Riemannian)L1 norm,outlinethatthis isnotcompleteonC0([0,1]).Briefdescriptionofl2 ThisisHilbertspace,butnotexplained.Whatisitallfor? Mainaims:- Tobeabletocarryoutstandardconstructionsin(linear)functionalanalysis:

AbstractHilbertspaceone ineachdimensionConcreteHilbertspaceMany,suchasL2([0,1])Example of a theorem:- The Dirichlet problem. Let V : [0,1] R be areal-valued function. We are interested in oscillating modes on the interval;somethinglikethisarisesinquantummechanicsforinstance. Namelywewanttoknowaboutfunctionsu(x)twicecontinuouslydifferentiableon[0,1]whichsatisfythedifferentialequation

d2u(1.1)

dx2(x) +V(x)u(x) =u(x)where is an unknown constant that is we want to know which s canoccur. Well, of course all s can occur with u 0 but this is the trivialsolutionwhichwillalwaysbethereforsuchanequation. Whatothersolutionsare there? Well, there is an infinite sequence of s for which there is a nontrivialsolutionof (1.1)j Rtheyareallrealnonon-realcomplexscanoccur. Foreachofthesethereisatleastone(andmaybemore)independentsolutionuj.Wecansayalotmoreabouteverythingherebutonemainaimofthiscourse istogetatleasttothispoint.

Now,infact(1.1)isjusttheeigenvalueequation. Whatwearedealingwithhereisan

infinite

matrix.

This

is

not

obvious,

and

in

fact

is

not

avery

good

way

of

lookingatthings(therewassuchamatrixapproachtoquantummechanicsintheearlydaysbut itwasreplacedbythesortof operatortheoryonHilbertspace that we will use here.) Still, we are in some sensedealing with infinitedimensionalmatrices. Oneofthecrucialdifferencesbetweeninfiniteandfinitedimensionalsettingsisthattopology isencountered. Thisisenshrinedhereinthenotionofanormed linearspace

Linearspace:- ShouldIbreakouttheaxioms? ItisaspaceV inwhichwecanaddelementsandmultiplybyscalarswithrulesverysimilartothebasicexamplesofRn orCn. Notethat forus the scalars are eitherthe realnumbersof thecomplexnumbersusuallythelatter. LetsbeneutralanddenotebyKeitherR orC but of course consistently. Then our set V the set of vectors withwhichwewilldeal,comeswithtwolaws. Thesearemaps

(1.2) +:V V V, :KV V.which we denote not by +(v, w) and (s, v) but by v+w and sv. Then weimposetheaxioms of a vector space lookthemup! Thesearecommutativegroupaxiomsfor+axiomsfortheactionofKandthedistributive law.

Ourexamples:Thetrivialcaseofafinitedimensionalvectorspace.The lp spaces. These are spaces of sequences the first problem set is allaboutthem. Thusl2 whichisaHilbertspaceconsistsofallthesequences

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5LECTURE NOTES FOR 18.102, SPRING 2009

a:NCalsodenoted{aj} whereaj =a(j),suchthatj=1

(1.3) |aj|2

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6 LECTURE NOTES FOR 18.102, SPRING 2009

Problemset1,Due11AMTuesday10Feb.Full marks will be given to anyone who makes a good faith attempt to answer

eachquestion. Thefirst fourproblemsconcernthe littleLpspaceslp.Notethatyouhavethechoiceofdoingeverythingforp= 2orforall1p