We present fundamental limits on the reliable classification of linear and affine subspaces from noisy, linear features. Drawing an analogy between discrimination among subspaces and communication over vector wireless channels, we propose two Shannon-inspired measures to characterize asymptotic classifier performance. First, we define the classification capacity, which characterizes necessary and sufficient conditions for the misclassification probability to vanish as the signal dimension, the number of features, and the number of subspaces to be discerned all approach infinity. Second, we define the diversity-discrimination tradeoff which, by analogy with the diversity-multiplexing tradeoff of fading vector channels, characterizes relationships between the number of discernible subspaces and the misclassification probability as the noise power approaches zero. We derive upper and lower bounds on these measures which are tight in many regimes. Numerical results, including a face recognition application, validate the results in practice.