  We discuss the superfluid properties of a Bose-Einstein condensed gas with
spin-orbit coupling, recently realized in experiments. We find a finite normal
fluid density $\rho_n$ at zero temperature which turns out to be a function of
the Raman coupling. In particular, the entire fluid becomes normal at the
transition point from the zero momentum to the plane wave phase, even though
the condensate fraction remains finite. We emphasize the crucial role played by
the gapped branch of the elementary excitations and discuss its contributions
to various sum rules. Finally, we prove that an independent definition of
superfluid density $\rho_s$, using the phase twist method, satisfies the
equality $\rho_n+\rho_s=\rho$, the total density, despite the breaking of
Galilean invariance.
