Answer to Question #12465 in Linear Algebra for john.george.milnor
Prove that set of all sequences that satisfy Hilbert condition form a linear subspace
It sufficient to show that sum of every two such sequences is again the same type. Let we have two convergent series Series(|a_n|^2) and Series(|b_n|^2). They can be regarded as norm of two elements of infinite dimensional normed space R^(inf). Thus as norm has property ||a+b||<=||a||+||b|| then Series(|a_n+b_n|^2) is convergent too, so sequences with property Series(|a_n|^2)<(inf) are subspace in R^(inf).