  For each $f\in L^p({\mathbb R)}$ ($1\leq p<\infty$) it is shown that the
Fourier transform is the distributional derivative of a H\"older continuous
function. For each $p$ a norm is defined so that the space Fourier transforms
is isometrically isomorphic to $L^p({\mathbb R)}$. There is an exchange theorem
and inversion in norm.
