In $R^n$ with the standard inner product, the Cauchy-Schwarz inequality is $$ \left ( \sum_{i=1}^n a_i b_i \right)^2 \le \left ( \sum_{i=1}^n a_i ^2\right) \left ( \sum_{i=1}^n b_i^2 \right) .$$
Is there an analog of Cauchy-Schwarz inequality for double summation? Say, how does one apply Cauchy-Schwarz to something like: $$ \sum_{i=1}^n\sum_{j=1}^n a_i b_j ~?$$