A Note on the Quasi-Best Approximation Constant

Abstract The solution operator in a Petrov Galerkin scheme in Hilbert spaces is an oblique projection with quasi-best approximation property. The latter estimate involves a multiplicative constant and the best-possible of those is the target of the note: We present a new direct proof of the formula of the quasi-best approximation constant and avoid the direct application of the Kato lemma. In fact, our characterisation leads to another proof of the Kato oblique projection lemma. The abstract result in Hilbert spaces is embedded in the setting of the best-approximation of conforming Petrov Galerkin schemes with a rich history that eventually led to the Tantardini–Veeser formula. A final application discusses the classical nonconforming schemes with a smoother in this framework.