Banach space projections and Petrov-Galerkin estimates

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.