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In this paper, training-based estimation of correlated block fading channels in a multiple-antenna, multi-user environment is considered. The linear minimum mean squared error (LMMSE) estimator is presented first. Then the problem of optimally designing the training data set, so as to minimize the mean squared channel estimation error subject to a total transmit power constraint, is addressed. It is shown that the optimal transmission directions are dictated jointly by the eigen-decompositions of the channel and interference covariance matrices. Their roles, in the channel estimation and interference suppression tasks, respectively, are revealed in the optimal transmit beamformer structure. The simulation results demonstrate that the gain in estimation performance from using the optimal training sequence increases considerably with increasing spatial fading correlation, especially in strong interference environments.