The norm and the Fourier coefficient if the inner product $\langle f(x), f(x)\rangle$ is negative.

Question

For a real function $f_m(x)$ orthogonal with respect to $w(x)$ where $w(x)>0$. We have inner product $$ \langle f_m,f_n\rangle_{w}=\int_{x_0}^{x_0 + T}{w(x)f_m(x)f_n(x)dx} $$ The norm is $||f_m||_{w}=\sqrt{\langle f_m,f_m\rangle_{w}}$, and the Fourier coefficient of function $g(x)$ is $a_m=\frac{\langle g,f_m\rangle_w}{||f_m||_w^2}$.

Here's the question: what if $\langle f_m,f_m\rangle_w<0$? Does the norm $||f_m||=i\sqrt{-\langle f_m,f_m\rangle_w}$ and the Fourier coefficient $a_m=\frac{\langle g,f_m\rangle_w}{||f_m||_w^2}$ still apply?

Thanks!