Question

Prove that set of all sequences that satisfy Hilbert condition form a linear subspace
in R^(inf)

Answer 1

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).