  The Barankin bound is generalized to the vector case in the mean square error
sense. Necessary and sufficient conditions are obtained to achieve the lower
bound. To obtain the result, a simple finite dimensional real vector valued
generalization of the Riesz representation theorem for Hilbert spaces is given.
The bound has the form of a linear matrix inequality where the covariances of
any unbiased estimator, if these exist, are lower bounded by matrices depending
only on the parametrized probability distributions.
