Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods

We give an elementary, coordinate-free proof that uniformly inf-sup stablenested Petrov--Galerkin methods on Hilbert spaces satisfy relaxed generalquasi-orthogonality. More precisely, the accumulated squared Galerkinincrements over any window of \(N\) consecutive levels are bounded by thesquared error at the beginning of the window times \(N^σ\), where \(σ<1\), and both \(σ\) and the constant prefactor depend explicitlyonly on a uniform bound for the Galerkin projections. The proof uses the Hilbert-space angle between consecutive blocks of auniformly bounded compatible projection chain, together with a dyadicdecomposition and duality.It avoids matrix representations, wavelet bases, and\(LU\)-factorization. For symmetric indefinite problems, we relate theargument to the positive and negative spectral splittings of the Galerkindetail spaces and obtain a valid finite-window version of thesign-decomposition approach. We also construct a fixed self-adjoint involution and a fixed nested,uniformly inf-sup stable Galerkin sequence for which full generalquasi-orthogonality fails. The same construction yields, for every\(0<α<1\), finite-support targets whose full-tail ratios grow at least like\(N^α\).Thus uniform inf-sup stability guaranteessublinear finite-window quasi-orthogonality, whereas fullquasi-orthogonality requires additional hierarchical information ingeneral.