  In this paper, we consider the error analysis of a conservative Fourier
pseudo-spectral method that conserves mass and energy for the space fractional
nonlinear Schr\"{o}dinger equation. We give a new fractional Sobolev norm that
can construct the discrete fractional Sobolev space, and we also can prove some
important lemmas for the new fractional Sobolev norm. Based on these lemmas and
energy method, a priori error estimate for the method can be established. Then,
we are able to prove that the Fourier pseudo-spectral method is unconditionally
convergent with order $O(\tau^{2}+N^{\alpha/2-r})$ in the discrete $L^{\infty}$
norm, where $\tau$ is the time step and $N$ is the number of collocation points
used in the spectral method. Numerical examples are presented to verify the
theoretical analysis.
