  In this paper we find the optimal error bound (smallest possible estimate,
independent of the starting point) for the linear convergence rate of the
simultaneous projection method applied to closed linear subspaces in a real
Hilbert space. We achieve this by computing the norm of an error operator which
we also express in terms of the Friedrichs number. We compare our estimate with
the optimal one provided for the alternating projection method by Kayalar and
Weinert (1988). Moreover, we relate our result to the alternating projection
formalization of Pierra (1984) in a product space. Finally, we adjust our
results to closed affine subspaces and put them in context with recent
dichotomy theorems.
