Channel estimation is very challenging when the receiver is equipped with a limited number of radio-frequency (RF) chains in beamspace millimeter-wave massive multiple-input and multiple-output systems. To solve this problem, we exploit a learned denoising-based approximate message passing (LDAMP) network. This neural network can learn channel structure and estimate channel from a large number of training data. Furthermore, we provide an analytical framework on the asymptotic performance of the channel estimator. Based on our analysis and simulation results, the LDAMP neural network significantly outperforms state-of-the-art compressed sensing-based algorithms even when the receiver is equipped with a small number of RF chains.