let X be RealUnitarySpace; :: thesis: ( the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity implies for Y being Subset of X st Y is weakly_summable_set holds
for L being linear-Functional of X st L is Bounded holds
Y is_summable_set_by L )
assume A1: ( the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity ) ; :: thesis: for Y being Subset of X st Y is weakly_summable_set holds
for L being linear-Functional of X st L is Bounded holds
Y is_summable_set_by L
let Y be Subset of X; :: thesis: ( Y is weakly_summable_set implies for L being linear-Functional of X st L is Bounded holds
Y is_summable_set_by L )
assume A2: Y is weakly_summable_set ; :: thesis: for L being linear-Functional of X st L is Bounded holds
Y is_summable_set_by L
let L be linear-Functional of X; :: thesis: ( L is Bounded implies Y is_summable_set_by L )
assume A3: L is Bounded ; :: thesis: Y is_summable_set_by L
consider x being Point of X such that
A4: for L being linear-Functional of X st L is Bounded holds
for e being Real st e > 0 holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs ((L . x) - (setopfunc Y1,the carrier of X,REAL ,L,addreal )) < e ) ) by A1, A2, Th6;
for e being Real st e > 0 holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs ((L . x) - (setopfunc Y1,the carrier of X,REAL ,L,addreal )) < e ) ) by A3, A4;
hence Y is_summable_set_by L by Def6; :: thesis: verum