All of the sequences, mentioned in this post, are real sequences. $l_1$ is the set of real sequences such that the sum of absolute value of the terms converges and $l_2$ is the set of real sequences such that sum of squares of the terms converges.
My question is: for which fixed $(\alpha_n)_{n=1}^{\infty}$, the mapping $(x_n)_{n = 1}^{\infty} \in l_2 \mapsto (\alpha_nx_n)_{n=1}^{\infty}\in l_1$ is well-defined. My guess is that the mapping is well-defined if and only if $(\alpha_n)_{n=1}^{\infty} \in l_2$. I prove one direction of the proposition as follows:
Assume $(\alpha_n)_{n=1}^{\infty} \in l_2$. By Hölder's Inequality, we have $\sum_{n= 1}^{\infty}|x_n|| \alpha_n| \le (\sum_{n= 1}^{\infty}x_n^2)^{\frac{1}{2}}(\sum_{n= 1}^{\infty}\alpha_n^2)^{\frac{1}{2}}< \infty$. I need help with the other direction of the proposition.
