Question #12465

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

Expert's answer

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).

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).

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