Proving Hilbert transform is skew-adjoint

Question

I want to solve the following exercise from Ioannis Parissis's lecture notes (corollary 5)

The Hilbert transform, $H$, defined for $f\in L^2(\mathbb{R})$ as $$\widehat{Hf}(\xi) = -i \: \text{sgn}(\xi)\hat{f}(\xi) $$ is skew-adjoint, i.e. for $f,g\in L^2(\mathbb{R})$ $$\int_{\mathbb{R}}Hf(x)\overline{g(x)}dx = -\int_{\mathbb{R}}f(x)\overline{Hg(x)}dx$$

But have no idea how to approach this problem. How can Fourier transform be used with integral?

Answer 1

$\langle f,g \rangle_{L^2} = \langle \hat{f}, \hat{g} \rangle_{L^2}$