Cauchy-Schwarz inequality in $L^2$

Question

In Cauchy-Schwarz Master Class, it states that Schwarz considered the polynomial $p(t)=\int\int_S (tf(x,y)+g(x,y))^2dxdy$ to show that if $f(x,y), g(x,y)$ are not proportional, then p(t) is strictly positive.

Without resorting to analysis and assuming that $f$ and $g$ are continuous on $S\subset\mathbb{R}^2$, I don't see why this is necessarily true. Is there something elementary that I'm missing?

Answer 1

$p(t)=0$ is equivalent to $g(x,y)=-tf(x,y)$ almost everywhere. We get equality for all $x,y$ when the functions are continuous. When they say proportional they mean $g(x,y)=cf(x,y)$ almost everywhere for a constant $c$.