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: $$ \sum_{i=1}^n\sum_{j=1}^n a_i b_j ~?$$
Best Answer
Rewrite the sum you are curious about as $$\Big( \sum_{i=1}^n a_i \Big) \Big( \sum_{j=1}^n b_j \Big)$$ Then, I'm not really sure what you want to use Cauchy-Schwarz for here, but you could say $$\Big( \sum_{i=1}^n a_i \Big)^2 \Big( \sum_{j=1}^n b_j \Big)^2 \leq n^2 \Big( \sum_{i=1}^n a_i^2 \Big) \Big( \sum_{j=1}^n b_j^2 \Big)$$