theorem :: BHSP_6:10

for X being RealHilbertSpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds
for Y being Subset of X holds
( Y is summable_set iff 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 not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
||.(setsum Y1).|| < e ) ) )