Hybrid analog/digital precoding is a promising technique to reduce the hardware cost of radio-frequency components compared with the conventional full-digital precoding approach in millimeter-wave multiple-input multiple output systems. However, the large antenna dimensions of the hybrid precoder design makes it difficult to acquire an optimal full-digital precoder. Moreover, it also requires matrix inversion, which leads to high complexity in the hybrid precoder design. In this paper, we propose a low-complexity optimal full-digital precoder acquisition algorithm, named beamspace singular value decomposition (SVD) that saves power for the base station and user equipment. We exploit reduced-dimension beamspace channel state information (CSI) given by compressive sensing (CS) based channel estimators. Then, we propose a CS-assisted beamspace hybrid precoding (CS-BHP) algorithm that leverages CS-based CSI. Simulation results show that the proposed beamspace-SVD reduces complexity by 99.4% compared with an optimal full-digital precoder acquisition using full-dimension SVD. Furthermore, the proposed CS-BHP reduces the complexity of the state-of-the-art approach by 99.6% and has less than 5% performance loss compared with an optimal full-digital precoder.