Similar to the question listed here: Fourier switching in $L^2(R)/S(R)$ in the integral
If $f \in L^1(\mathbb{R}), g \in S(\mathbb{R})$, (where $S(\mathbb{R})$ is the Schwartz space) then how can I show that, integrating over $\mathbb{R}$: $$ \int \hat{f}(\xi) \, g(\xi) \, d\xi = \int f(\xi) \, \hat{g}(\xi) \, d\xi $$
I know that the fourier switching works for $f,g \in S(\mathbb{R})$. So I wanted to use the continuity of the fourier transformation and the schwartz space's density to extend to $L^1$, but the fourier transformation does not necessarily map to $L^1$. What is the best way to prove this?