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1、SpacesofRandomVariablesPeterOuwehandDepartmentofMathematicalSciencesUniversityofStellenboschNovember2010P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20101/11TopologicalVectorSpacesIDe nition:Anormedspaceisapair(V;jjjj),whereVisavectorspaceandjjjjisanormonV,i.e.a
2、functionjjjj:V!Rwiththefollowingproperties:P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDe nition:Anormedspaceisapair(V;jjjj),whereVisavectorspaceandjjjjisanormonV,i.e.afunctionjjjj:V!Rwiththefollowingproperties:(i)jjxjj0forall
3、x2V;P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDe nition:Anormedspaceisapair(V;jjjj),whereVisavectorspaceandjjjjisanormonV,i.e.afunctionjjjj:V!Rwiththefollowingproperties:(i)jjxjj0forallx2V;(ii)jjxjj=0ifandonlyifx=0;P.Ouwehand
4、(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDe nition:Anormedspaceisapair(V;jjjj),whereVisavectorspaceandjjjjisanormonV,i.e.afunctionjjjj:V!Rwiththefollowingproperties:(i)jjxjj0forallx2V;(ii)jjxjj=0ifandonlyifx=0;(iii)jjxjj=jjjjxjjforall
5、x2Vand2R;P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDe nition:Anormedspaceisapair(V;jjjj),whereVisavectorspaceandjjjjisanormonV,i.e.afunctionjjjj:V!Rwiththefollowingproperties:(i)jjxjj0forallx2V;(ii)jjxjj=0ifandonlyifx=0;(iii)
6、jjxjj=jjjjxjjforallx2Vand2R;(iv)jjx+yjjjjxjj+jjyjjforallx;y2V({Inequality);P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDe nition:Anormedspaceisapair(V;jjjj),whereVisavectorspaceandjjjjisanormonV,i.e.afunctionjjjj:V!Rwiththefol
7、lowingproperties:(i)jjxjj0forallx2V;(ii)jjxjj=0ifandonlyifx=0;(iii)jjxjj=jjjjxjjforallx2Vand2R;(iv)jjx+yjjjjxjj+jjyjjforallx;y2V({Inequality);Thinkofjjvjjasthelengthofv.P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDe nition:Anorm
8、edspaceisapair(V;jjjj),whereVisavectorspaceand
