If the "squared" infinite series $\sum_{n=0}^\infty a_n^2$ and $\sum_{n=0}^\infty b_n^2$ are both convergent then the "product" infinite series $\sum_{n=0}^\infty a_n b_n$ is absolutely convergent, and the following generalized Cauchy-Schwarz inequality holds:
$$\left|\sum_{n=0}^\infty a_n b_n\right|\le \sum_{n=0}^\infty |a_n b_n|\le \left(\sum_{n=0}^\infty a_n^2 \right)^{\frac 12}\left(\sum_{n=0}^\infty b_n^2 \right)^{\frac 12}.$$
Proofs: 1
