Fourier transform has proper dense image from L1 to C0
= Fourier transform has proper dense image from L1 to C0
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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 $\chi_{[-n,n]}*\chi_{[-1,1]}$ have unbounded $L^1$ norms while the targets have <uniform norm> $2$, contradicting the bounded inverse that the <open mapping theorem> would give.