  We sharpen the classic a priori error estimate of Babuska for Petrov-Galerkin
methods on a Banach space. In particular, we do so by (i) introducing a new
constant, called the Banach-Mazur constant, to describe the geometry of a
normed vector space; (ii) showing that, for a nontrivial projection $P$, it is
possible to use the Banach-Mazur constant to improve upon the naive estimate $
\| I - P \| \leq 1 + \| P \| $; and (iii) applying that improved estimate to
the Petrov-Galerkin projection operator. This generalizes and extends a 2003
result of Xu and Zikatanov for the special case of Hilbert spaces.
