inner product spaces

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1、September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch04page195Chapter4InnerProductSpaces4.1InnerProductDefinition4.1.1LetF=R,CandletVbeavectorspaceoverF.Then
·,·:V×V→Fiscalledaninnerproductifthefollowingconditionshold:(a)
ax+by,z=a
x,z

2、+b
y,z,foralla,b∈F,x,y,z∈V,(br)forF=R
y,x=
x,y,forallx,y∈V;(bc)forF=C
y,x=
x,y,forallx,y∈V;(c)
x,x>0forallx∈V{0}.
x:=
x,xiscalledthenorm(length)ofx∈V.OtherstandardpropertiesofinnerproductsarementionedinProblems1and2.WewillusetheabbreviationIPSfori

3、nnerprod-uctspace.Inthischapter,weassumethatF=R,Cunlessstatedotherwise.Proposition4.1.2LetVbeavectorspaceoverR.IdentifyVC,calledthecomplexificationofV,withthesetofpairs(x,y),195September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch04page1

4、96196Matricesx,y∈V.ThenVCisavectorspaceoverCwith√(a+−1b)(x,y):=a(x,y)+b(−y,x),foralla,b∈R,x,y∈V.IfVhasabasise1,...,enoverRthen(e1,0),...,(en,0)isabasisofVCoverC.Anyinnerproduct
·,·onVoverFinducesthefollowinginnerproductonVC:√
(x,y),(u,v)=
x,u+
y,v+−1(
y,

5、u−
x,v),x,y,u,v∈V.Weleavetheproofofthispropositiontothereader(Problem3).Definition4.1.3LetVbeanIPS.Then(a)x,y∈Varecalledorthogonalif
x,y=0.(b)S,T⊂Varecalledorthogonalif
x,y=0foranyx∈S,y∈T.(c)ForanyS⊂V,S⊥⊂VisthemaximalorthogonalsettoS.(d)x1,...,xmiscalleda

6、northonormalsetif
xi,xj=δij,foralli,j=1,...,m.(e)x1,...,xniscalledanorthonormalbasisifitisanorthonormalsetwhichisabasisinV.Definition4.1.4(Gram–Schmidtalgorithm)LetVbeanIPSandS={x1,...,xm}⊂Vafinite(possiblyempty)set(m≥0).ThenS˜={e1,...,ep}istheorthonormalset(

7、p≥1)ortheemptyset(p=0)obtainedfromSusingthefollowingrecursivesteps:(a)Ifx1=0removeitfromS.Otherwisereplacex1byx1−1x1.(b)Assumethatx1,...,xkisanorthonormalsetand1≤k

8、eptember21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch04page197InnerProductSpaces197Corollary4.1.5LetVbeanIPSandS={x1,...,xn}⊂Vbenlinearlyindependentvectors.ThentheGram–Schmi