  This paper aims first at a simultaneous axiomatic presentation of the proof
of optimal convergence rates for adaptive finite element methods and second at
some refinements of particular questions like the avoidance of (discrete) lower
bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent
error estimators. Solely four axioms guarantee the optimality in terms of the
error estimators.
  Compared to the state of the art in the temporary literature, the
improvements of this article can be summarized as follows: First, a general
framework is presented which covers the existing literature on optimality of
adaptive schemes. The abstract analysis covers linear as well as nonlinear
problems and is independent of the underlying finite element or boundary
element method. Second, efficiency of the error estimator is neither needed to
prove convergence nor quasi-optimal convergence behavior of the error
estimator. In this paper, efficiency exclusively characterizes the
approximation classes involved in terms of the best-approximation error and
data resolution and so the upper bound on the optimal marking parameters does
not depend on the efficiency constant. Third, some general quasi-Galerkin
orthogonality is not only sufficient, but also necessary for the $R$-linear
convergence of the error estimator, which is a fundamental ingredient in the
current quasi-optimality analysis due to Stevenson 2007. Finally, the general
analysis allows for equivalent error estimators and inexact solvers as well as
different non-homogeneous and mixed boundary conditions.
