Lecture_Spaces_Random_Variables

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1、SpacesofRandomVariablesPeterOuwehandDepartmentofMathematicalSciencesUniversityofStellenboschNovember2010P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20101/11TopologicalVectorSpacesIDenition:Anormedspaceisapair(V;jj
jj),whereVisavectorspaceandjj
jjisanormonV,i.e.a

2、functionjj
jj:V!Rwiththefollowingproperties:P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDenition:Anormedspaceisapair(V;jj
jj),whereVisavectorspaceandjj
jjisanormonV,i.e.afunctionjj
jj:V!Rwiththefollowingproperties:(i)jjxjj0forall

3、x2V;P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDenition:Anormedspaceisapair(V;jj
jj),whereVisavectorspaceandjj
jjisanormonV,i.e.afunctionjj
jj:V!Rwiththefollowingproperties:(i)jjxjj0forallx2V;(ii)jjxjj=0ifandonlyifx=0;P.Ouwehand

4、(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDenition:Anormedspaceisapair(V;jj
jj),whereVisavectorspaceandjj
jjisanormonV,i.e.afunctionjj
jj:V!Rwiththefollowingproperties:(i)jjxjj0forallx2V;(ii)jjxjj=0ifandonlyifx=0;(iii)jjxjj=jjjjxjjforall

5、x2Vand2R;P.Ouwehand(StellenboschUniv.)SpacesofRandomVariablesNovember20102/11TopologicalVectorSpacesIDenition:Anormedspaceisapair(V;jj
jj),whereVisavectorspaceandjj
jjisanormonV,i.e.afunctionjj
jj: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;jj
jj),whereVisavectorspaceandjj
jjisanormonV,i.e.afunctionjj
jj: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;jj
jj),whereVisavectorspaceand