  Based on the NMR measurements on Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$
(La-Bi2201) in strong magnetic fields, we identify the non-superconducting
pseudogap phase in the cuprates as a Luttinger-volume violating Fermi liquid
(LvvFL). This state is a zero temperature quantum liquid that does not break
translational symmetry, and yet, the Fermi surface encloses a volume smaller
than the large one given by the Luttinger theorem. The particle number enclosed
by the small Fermi surface in the LvvFL equals the doping level $p$, not the
total electron number $n_e=1-p$. Both the phase string theory and the dopon
theory are introduced to describe the LvvFL. For the dopon theory, we can
obtain a semi-quantitative agreement with the NMR experiments.
