 Fourier transform has proper dense image from L1 to C0

The Riemann-Lebesgue lemma and Fourier inversion give a bounded injective map $L^1(\mathbb{R})\to C_0(\mathbb{R})$ with dense image. It is not onto: inverse transforms of ${\chi }_{[-n,n]}\ast {\chi }_{[-1,1]}$ have unbounded $L^1$ norms while the targets have uniform norm $2$, contradicting the bounded inverse that the open mapping theorem would give.