Proving the discrete version of Cauchy-Schwarz is easy:
$ \left(\sum_{i} a_i^2\right) \left(\sum_i b_i^2\right) \geq \left(\sum_i a_ib_i\right)^2 $
can be done via the determinant of the quadratic formula.
Now, however, I want to prove the continuous version, which states:
$ \int a^2 \int b^2 \geq \left(\int ab\right)^2$
How do I prove this?