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In this paper, we investigate spatial multiplexing at millimeter (mm) wave carrier frequencies for short-range indoor applications by quantifying fundamental limits in line-of-sight (LOS) environments and then investigating performance in the presence of multipath and LOS blockage. Our contributions are summarized as follows. For linear arrays with constrained form factor, an asymptotic analysis based on the properties of prolate spheroidal wave functions shows that a sparse array producing a spatially uncorrelated channel matrix effectively provides the maximum number of spatial degrees of freedom in a LOS environment, although substantial beamforming gains can be obtained by using denser arrays. This motivates our proposed mm-wave MIMO architecture, which utilizes arrays of subarrays to provide both directivity and spatial multiplexing gains. System performance is evaluated in a simulated indoor environment using a ray-tracing model that incorporates multipath effects and potential LOS blockage. Eigenmode transmission with waterfilling power allocation serves as a performance benchmark, and is compared to the simpler scheme of beamsteering transmission with MMSE reception and a fixed signal constellation. Our numerical results provide insight into the spatial variations of attainable capacity within a room, and the combinations of beamsteering and spatial multiplexing used in different scenarios.