  We study the alternating algorithm for the computation of the metric
projection onto the closed sum of two closed subspaces in uniformly convex and
uniformly smooth Banach spaces. For Banach spaces which are convex and smooth
of power type, we exhibit a condition which implies linear convergence of this
method. We show these convergence results for iterates of Bregman projections
onto closed linear subspaces. Using an intimate connection between the metric
projection onto a closed linear subspace and the Bregman projection onto its
annihilator, we deduce the convergence rate results for the alternating
algorithm from the corresponding results for the iterated Bregman projection
method.
