Suppose that the functions are integrable on the closed, bounded interval [a,b]. Prove that:
then the previous equation becomes
How do I get rid of the 2 at the bottom?
The Cauchy-Schwarz inequality is a general inequality for inner product spaces.
Thus, what you wrote is a consequence of this general inequality found here*.
*)I am just a little worried because the space of all integral functions is not an inner product space with norm .
However, the proof on that page seems to work.
A more general view on this inequality is Hölder's inequality.
Cauchy-Schwarz is then a specific situation, with p=q=2.
Hölder's inequality - Wikipedia, the free encyclopedia