  We prove new optimality results for adaptive mesh refinement algorithms for
non-symmetric, indefinite, and time-dependent problems by proposing a
generalization of quasi-orthogonality which follows directly from the inf-sup
stability of the underlying problem. This completely removes a central
technical difficulty in modern proofs of optimal convergence of adaptive mesh
refinement algorithms and leads to simple optimality proofs for the Taylor-Hood
discretization of the stationary Stokes problem, a
finite-element/boundary-element discretization of an unbounded transmission
problem, and an adaptive time-stepping scheme for parabolic equations. The main
technical tool are new stability bounds for the LU-factorization of matrices
together with a recently established connection between quasi-orthogonality and
matrix factorization.
