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

Banach space projections and Petrov-Galerkin estimates