  This work presents a comprehensive discretization theory for abstract linear
operator equations in Banach spaces. The fundamental starting point of the
theory is the idea of residual minimization in dual norms, and its inexact
version using discrete dual norms. It is shown that this development, in the
case of strictly-convex reflexive Banach spaces with strictly-convex dual,
gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently,
abstract mixed methods with monotone nonlinearity. Crucial in the formulation
of these methods is the (nonlinear) bijective duality map.
  Under the Fortin condition, we prove discrete stability of the abstract
inexact method, and subsequently carry out a complete error analysis. As part
of our analysis, we prove new bounds for best-approximation projectors, which
involve constants depending on the geometry of the underlying Banach space. The
theory generalizes and extends the classical Petrov-Galerkin method as well as
existing residual-minimization approaches, such as the discontinuous
Petrov-Galerkin method.
