  We give the first rigorous (non perturbative) proof of Luttinger liquid
behavior in the one dimensional Hubbard model, for small repulsive interaction
and values of the density different from half filling. The analysis is based on
the combination of multiscale analysis with Ward identities bases on a hidden
and approximate local chiral gauge invariance. No use is done of exact
solutions or special integrability properties of the Hubbard model, and the
results can be in fact easily generalized to include non local interactions,
magnetic fields or interaction with external potentials
