Entropy function spaces and interpolation

We associate to every function space, and to every entropy function E, a scale of spaces Λp,q(E) similar to the classical Lorentz spaces Lp,q. Necessary and sufficient conditions for they to be normed spaces are proved, their role in real interpolation theory is analyzed, and a number of applications to functional and interpolation properties of several variants of Lorentz spaces and entropy spaces are given.