Proving Parseval Identity $\int_{\mathbb{R}^N}f(x)\overline{g(x)}dx = \frac{1}{(2\pi)^N}\int_{\mathbb{R}^N}\hat{f}(\eta)\overline{\hat{g}(\eta)}d\eta$

Question

Prove the Parseval Identity:

$$\int_{\mathbb{R}^N}f(x)\overline{g(x)}dx = \frac{1}{(2\pi)^N}\int_{\mathbb{R}^N}\hat{f}(\eta)\overline{\hat{g}(\eta)}d\eta, \ f,g\in S(\mathbb{R}^N)$$

Where $\hat{f}$ is the fourier transform of $f$. I don't know what $\overline{g}$ means. At least it's not in the class notes I took. I think it's crucial to understand how to prove it.

Note: $S$ is the Schrwartz space

Answer 1

Observe \begin{align} \int dx\ f(x)\overline{g(x)} =&\ \int dxdy\ f(x)\overline{g(y)}\delta(x-y)\\ =&\ \int dxdyd\xi\ f(x)\overline{g(y)} e^{-2\pi i(x-y)\xi}\\ =&\ \int d\xi \left(\int dx\ f(x)e^{-2\pi i x\cdot \xi} \right)\overline{\left(\int dy\ g(y)e^{-2\pi i y\cdot \xi} \right)}\\ =&\ \int d\xi\ \hat f(\xi)\overline{\hat g(\xi)}. \end{align}

Note that I used the definition \begin{align} \hat f(\xi) = \int dx\ f(x) e^{-2\pi i x \cdot \xi}. \end{align}