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Millimeter-wave (mmWave) massive multiple-input-multiple-output (mMIMO) is reported as a key enabler in the fifth-generation communication and beyond. It is customary to use a lens antenna array to transform a mmWave mMIMO channel into a beamspace where the channel exhibits sparsity. This beamspace transformation is equivalent to performing a Fourier transformation of the channel. Still, a Fourier transformation is not necessarily the optimal one, due to many reasons. Accordingly, this paper proposes using a learned sparsifying dictionary as the transformation operator leading to another beamspace. Since the dictionary is obtained by training over actual channel measurements, this transformation is shown to yield two immediate advantages. First is enhancing channel sparsity, thereby leading to more efficient pilot reduction. Second is improving the channel representation quality, and thus reducing the underlying power leakage phenomenon. Consequently, this allows for both improved channel estimation and facilitated beam selection in mmWave mMIMO. Besides, a learned dictionary is also used as the precoding operator for the same reasons. Extensive simulations under various operating scenarios and environments validate the added benefits of using learned dictionaries in improving the channel estimation quality and the beam selectivity, thereby improving the spectral efficiency.