Banach space projections and Petrov–Galerkin estimates

We sharpen the classic a priori error estimate of Babuška for Petrov–Galerkin methods on a Banach space. In particular, we do so by (1) introducing a new constant, called the Banach–Mazur constant, to describe the geometry of a normed vector space; (2) showing that, for a nontrivial projection P, it is possible to use the Banach–Mazur constant to improve upon the naïve estimate (Formula presented.); and (3) 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.