We consider a spatially correlated multi-user communication system with Nt antennas at the base station (BS), U users (Nt of whom are serviced), and each user with a single antenna. While the users are assumed to know their channels perfectly, the BS has neither instantaneous channel state information (CSI) nor the spatial statistics corresponding to any of the users. We consider a limited feedback model where each user conveys B bits of partial channel information to the BS. Due to the lack of statistical information at the BS, a codebook based on random vector quantization (RVQ) and tailored to uncorrelated Rayleigh fading is used in signaling. Our first contribution is to characterize the impact of spatial correlation (in particular, the eigenvalues of the transmit covariance matrix) on the performance of an RVQ scheme. We also show that spatial correlation has significant consequences on practical aspects of limited feedback systems like storage and construction of codebooks. While independent RVQ codebooks have to be generated for the different users in the i.i.d. case, under certain relaxed assumptions on the correlation, the same codebook can be reused across all the users. This is because the spatial signature induced by the correlation can be exploited to distinguish the quantizers of different users.