Series convergence in $L_2$

Question

Is it true that if $\sum_{n=1}^{\infty} s_n^2$ converges and $u_n \in L_2[a;b]$ are bounded then the series $\sum_{n=1}^{\infty} s_n u_n$ converges to $L_2$ function? If so, how can I prove it?

Answer 1

It does hold, using Pythagores’ theorem and completeness, if the $u_n$ are pairwise orthogonal.

It is false in general though, take for instance $s_n=n^{-1}$ and $u_n=1$.