Millimeter-wave (mm-Wave) cellular systems are a promising option for a very high data rate communication because of the large bandwidth available at mm-Wave frequencies. Due to the large path-loss exponent in the mm-Wave range of the spectrum, directional beamforming with a large antenna gain is necessary at the transmitter, the receiver or both for capturing sufficient signal power. This in turn implies that fast and robust channel estimation plays a central role in systems performance since without a reliable estimate of the channel state the received signal-to-noise ratio (SNR) would be much lower than the minimum necessary for a reliable communication. In this paper, we mainly focus on single-antenna users and a multi-antenna base-station. We propose an adaptive sampling scheme to speed up the user's signal subspace estimation. In our scheme, the beamforming vector for taking every new sample is adaptively selected based on all the previous beamforming vectors and the resulting output observations. We apply the theory of optimal design of experiments in statistics to design an adaptive algorithm for estimating the signal subspace of each user. The resulting subspace estimates for different users can be exploited to efficiently communicate to the users and to manage the interference. We cast our proposed algorithm as low-complexity optimization problems, and illustrate its efficiency via numerical simulations.