Complex series and sequences

Question

Question: Let $\ell_2(\mathbb{C})$ be the set of all complex sequences $(z_n)\in\mathbb{C}$ such that $\sum_{n=0}^{\infty} |z_n|^2$ is convergent, i.e., $$\ell_2(\mathbb{C}) = \left\{ (z_n)_n\;:\; \sum_{n=0}^{\infty} |z_n|^2 \;\;\text{is convergent}\right\}.$$ If $(z_n),(w_n)\in\ell_2(\mathbb{C})$, show that $\displaystyle{\sum_{n=0}^{\infty} z_n\overline{w}_n}$ is convergent.

My attempt: Let $(z_n),(w_n)\in\ell_2(\mathbb{C})$. It follows from the Schwarz inequality that $$\left|\sum_{n=0}^{\infty} z_n\overline{w}_n\right|^2 \leq \left(\sum_{n=0}^{\infty} |z_n|^2\right)\left(\sum_{n=0}^{\infty} |w_n|^2\right).$$ Since $(z_n),(w_n)\in\ell_2(\mathbb{C})$, then the both series $\displaystyle{\sum_{n=0}^{\infty} |z_n|^2}$ and $\displaystyle{\sum_{n=0}^{\infty} |w_n|^2}$ converge and, therefore, the product $\displaystyle{\left(\sum_{n=0}^{\infty} |z_n|^2\right)\left(\sum_{n=0}^{\infty} |w_n|^2\right)}$ converges. Since such a product converges and the above inequality holds, then we can conclude from the comparison test that $\displaystyle{\sum_{n=0}^{\infty} z_n\overline{w}_n}$ is convergent.

Answer 1

Your attempt is okay, assuming that you can use the intended version of the Cauchy-Schwarz inequality.

Alternatively, you can work with the partial sums and then take the limit.

The infinite sum in the conclusion defines an inner product of the two elements in $\ell_2$.