I am learning of abstract Fourier analysis for the first time and I am having trouble seeing the Fourier inversion formula. It seems like it should not be so hard to prove.
Let $f \in L^1(G)$ where $G$ is locally compact abelian group and consider it's Haar measure. Then we define the Fourier transform of $f$ as: $$\hat{f}(\chi)=\int_{G}f(g)\overline{\chi(g)}$$ Now the Fourier inversion is: $$f(g)=\int_{\hat{G}}\hat{f}(\chi)\chi(g)$$ where the integral is over the haar measure of $\hat{G}$. I feel like this formula should be easy to prove somehow using Fubini and "Haar"-ness. But I was not able to figure it out.
As a side note: what does it mean that $\hat{f}$ vanishes over infinity? $G$ might not have a norm so what does infinity mean in this context?