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Lecture1:VectorSpacesandVectorNorms23456789(f)1111314151617119Oneofthemostremarkablefeaturesofvectorspacesisthenotionofdimension. Weneedonesimpleresultthatmakesthishappen,thebasistheorem.212223Normsareawayofputtingameasureofdistanceonvectorspaces. Thepurposeisfortherefinedanalysisofvectorspacesfromtheviewpointofmanyapplications.Itisalsotoallthecomparisonofvariousvectorsonthebasisoftheirlength.NormsderivedfrominnerproductIf,wecanproofthatitisavectornorm. Firstly,wewillprooftheThen,weprooftheInfact,(i)isanormderivedfromthestandardinnerproduct:29=3132343536Basicconcepts: vectorspace,subspace,span, linearcombination,linearlyindependent,lineardependent,basis,dimension, vectornorm,innerproduct.HOMEWORK