I have difficulty in proving Plancherel's formula in Fourier transform. Here is what I have thought:
In this question, I denote complex conjugation by an overline and (inverse) Fourier transform is defnined as
\begin{align} \mathcal{F}[f](\omega)&=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt, \\ \mathcal{F}^{-1}[F](t)&=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(t)e^{i\omega t}\,d\omega. \end{align}
Let $F=\mathcal{F}[f]$ and $G=\mathcal{F}[g]$.
\begin{align} \int_{-\infty}^{\infty}f(t)\overline{g(t)}\,dt&=\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega\right)\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}\overline{G(-\omega')}e^{i\omega' t}\,d\omega'\right)\,dt\\ &=\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega\right)\left(-\frac{1}{2\pi}\int_{-\infty}^{\infty}\overline{G(\omega')}e^{-i\omega' t}\,d\omega'\right)\,dt%\quad\text{(let $\omega''=-\omega'$ and re-write $\omega''$ to $\omega'$)} \\ &=-\left(\frac{1}{2\pi}\right)^2\int_{-\infty}^{\infty}F(\omega)\int_{-\infty}^{\infty}\overline{G(\omega')} \int_{-\infty}^{\infty}e^{i(\omega-\omega')t}\,dt\,d\omega'\,d\omega \\ &=-\left(\frac{1}{2\pi}\right)^2\int_{-\infty}^{\infty}F(\omega)\int_{-\infty}^{\infty}\overline{G(\omega')}2\pi\delta(\omega-\omega')\,d\omega'\,d\omega \\ &=-\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)\overline{G(\omega)}\,d\omega. \end{align}
So, an undesirable negative sign remains. Where do I make mistakes?
$$ \overline{g(t)} = \frac{1}{2\pi} \overline{\int_{-\infty}^\infty G(\omega) e^{i \omega t}\, d \omega } = \frac{1}{2\pi} \int_{-\infty}^\infty \overline{G(\omega)} e^{-i \omega t} \, d \omega $$