  Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is
defined on a Banach space $X$. In general, for solving this equation
numerically, a finite dimensional approximation of $X$ and an approximation of
$F$ are required. Moreover, in general the given data $\yd$ of $y$ are noisy.
In this paper we analyze finite dimensional variational regularization, which
takes into account operator approximations and noisy data: We show
(semi-)convergence of the regularized solution of the finite dimensional
problems and establish convergence rates in terms of Bregman distances under
appropriate sourcewise representation of a solution of the equation. The more
involved case of regularization in nonseparable Banach spaces is discussed in
detail. In particular we consider the space of finite total variation
functions, the space of functions of finite bounded deformation, and the
$L^\infty$--space.
