Banach space projections and Petrov-Galerkin estimates (Q2340900)

The author derives some estimate for the projection operator \(P\) on a Banach space, sharper than the naive estimate \(\|I-P\|\leq 1+\|P\|\). This is achieved by using the so-called Banach-Mazur constant to describe the geometry. Then this estimate is applied to improve the estimates for Petrov-Galerkin methods on a Banach space. This result generalizes an earlier work of \textit{J. Xu} and \textit{L. Zikatanov} [ibid. 94, No. 1, 195--202 (2003; Zbl 1028.65115)] in the context of a Hilbert space.