Let $G$ be a locally compact abelian group, then the Riemann-Lebesgue Lemma states that for a function $f\in L^1(G)$, then the Fourier transform $\hat{f}\in \mathcal{C}_0(\hat{G})$.
Here local compactness means that every neighbourhood of a point $x\in G$ has a compact sub-neighbourhood. We also assume that $G$ is Hausdorff.
All the proofs I've seen for this use Gelfand theory, but I'm wondering is there a way of completing this proof directly?