 Solution

Assume real square-integrable functions on the same integration domain. Put $A=\int f^2$, $B=\int g^2$, and $C=\int fg$. If $B=0$, then $g=0$ almost everywhere, so $C=0$ and the claim follows. If $B>0$, nonnegativity of a square givesMultiplying by $B$ proves the integral form of the Cauchy-Schwarz inequality:For nonzero functions, equality holds exactly when they are proportional almost everywhere. The zero-function cases also give equality.