My notes state that if we have two functions, $f$ and $g$, defined over the same domain, $\Omega $, then the inner product is the quantity:
$$ \langle{f,g}\rangle =\int_{\Omega}f(x)g(x)dx $$
One of the properties stated is that $ \langle{f,g}\rangle > 0$ for all non zero functions. However, in my eyes if on function is continuously negative, then $\langle{f,g}\rangle<0$. Please explain.
There is nothing wrong with $\langle f, g \rangle < 0$ , this is NOT an axiom. The axiom says, $\langle f, f \rangle \ge 0$ with equality if and only if $f=0$.