  We consider ill-posed linear operator equations with operators acting between
Banach spaces. For solution approximation, the methods of choice here are
projection methods onto finite dimensional subspaces, thus extending existing
results from Hilbert space settings. More precisely, general projection
methods, the least squares method and the least error method are analyzed. In
order to appropriately choose the dimension of the subspace, we consider a
priori and a posteriori choices by the discrepancy principle and by the
monotone error rule. Analytical considerations and numerical tests are provided
for a collocation method applied to a Volterra integral equation in one space
dimension.
