Let $\langle \cdot , \cdot \rangle : \mathbb{C}_{c}^{\infty}(\mathbb{R}) \times \mathbb{C}_{c}^{\infty}(\mathbb{R}) \rightarrow \mathbb{R}$ be defined as $$\langle f,g\rangle := - \int_{\mathbb{R}} \int_{\mathbb{R}} f(x) g(y) |x-y| dx dy$$
Show that the function ⟨·,·⟩ above defines an inner product over functions in $\mathbb{C}_{c}^{\infty}(\mathbb{R})$ that integrate to zero over $\mathbb{R}$.
I tried to simply the integral as $$\langle f,g\rangle := - \int_{\mathbb{R}} f(x)\ ( g(x) *|x|)\ dx$$
I am not able prove the properties of the inner product. Can someone please help me out by giving a small hint?
Use Parseval-Plancherel and the convolution theorem to express $\langle f,g\rangle $ in term of $\hat{f},\hat{g}$, find the Fourier transform of $|x|$ (its second derivative is $2\delta$), use the analyticity of $\hat{f}\hat{g}$ and that it has a double zero to obtain a simple formula for $\langle f,g\rangle $ making it obvious that it is an inner product.